To your salient points- I recently went on a 4 hour trip to the northern reaches of Maine where we take along a small boat and some kayaks. The boat trailer gets used often but primarily short, local trips. We dragged it to this same place last year. Two days before departure, I got to thinking the same thoughts you expressed. So I popped off one of the bearing caps to water coming out and milky grease. Off to the store for new bearing sets and marine grease. Few hours later wrapping up the bearing replacements and look at the tires. Sheesh, the tire that sits exposed to the sun has cracking around the circumference of the tread. So off to Tractor Supply for new wheel/tire combos. Lots of last minute preparation but major peace of mind. The hubs never even got hot at 75-80 mph for hours…and no breakdowns having to leave trailer on side of expressway = priceless.
When I had to use the trailer for an 800 mile one way trip to the kids, yeah I just replaced everything. But I just get the new hubs and bearings, plus tires. Plus I carried a spare hub and tire. I had a hub go bad on the camper once. A loaded and greased hub can be replaced pretty fast even on the road.
If you want to try yourself, the safest is to take the tire off and go somewhere that has air. However, it can still be done with the tire on the car, just be ready to scoot over to a tire store if there is an issue. Take the soap solution, replacement cores and core tool. You can try tightening but doubtful that will help. Before you remove the old core, prepare by over inflating the tire (not insanely but to somewhat compensate for the time the stem will be wide open). Remove the old core and quickly begin installing the new one. Any debris will be blown out when the core is removed. Seat the new core and refill the tire. Check with soap solution. If no bubbles, great. If still bubbling, head to the nearest tire store.
From what you’ve written, the air is leaking from the valve stem end. Two ways to check is to “spit on your finger” and rub the spit over the open valve end so that it forms a seal and then look for the seal to bubble. That it the old mechanics way.
But from your writings, I would suggest you get a small container and put some water and dish soap in it and create a bit of a lather then apply it to the value end and the valve base and look for bubbles, if the only bubbles that appear are at the end of the value stem only then you probably only need to tighten the valve core.
If so, go pick up this little tool from almost any auto parts store (even hardware stores…), and tighten the value core… YouTube has videos…
It goes by many names… Value Stem Tool:… some call it a 4-Way Valve Tool, or Tire Valve Tool, or 4-Way Valve Core Tool, or 4-in-1 Multifunctional Valve Wrench…
Also, if you do not have a good tire pressure gauge, pick one up too. If you need to add air while you are on your trip, the gauge will ensure you put in the correct pressure (see the inside of the driver’s side door post for proper pressure…).
About 3-years ago, we went to an Office Max to buy Rubber Bands and the associate had no idea what we were talking about, said he’d never heard of them… Got the manager and he found the Rubber Bands (even said it on the package…) and the associate said we should have asked for “Elastic Bands…”
I practically shoved the package in his face and said I would have if the package had called them “Elastic Bands…” I then turned to the manager and said, cant get good help, huh?
That works. They had the bearing set for $35, the loaded hubs were $100 ea. Done enough bearing jobs it doesn’t take me long plus I like to use marine grease for the boat trailer and the loaded hubs are packed with generic bearing grease. That must have been fun when the hub failed. When I was a kid, we had the tongue break and that was exciting for everyone in the car. Imagine how lucky, in front of a farm where the farmer welded it back together for my Dad.
Okay. So it sounds like replacing the core is not generally considered a safe and reliable repair. 
Bummer, but I guess that answers my question. We will cancel the trip.
It was no problem. Didn’t notice until home. The surface for the race just got elongated somehow. Maybe a bad bearing caused it and chewed the hub up. I think the whole pair of hubs, bearings, bolts, and dust caps was $35 back then from pioneer hub. Interesting waiting in the warehouse for my order to be picked with all the axles etc. being sold. Seemed like most orders were for the whole dang loaded axles with springs and hubs.
How did you come to that conclusion?
If the valve core is the problem, simple, easy to do.
Tool plus four cores less than $5.
If you have a good relationship with the shop that does your other maintenance, who knows, might have done it free.
I don’t know how you came to that conclusion? If you have bad tires that’s one thing, but a simple core replacement is like changing a light bulb. If you have properly identified that as the leak with spit or Soapy water, it’s a pretty simple thing to do. Ya goes down and buys the little box of cores and caps that has the tool in it, then park by an air hose and have at it. Oh yeah wear eye and ear protection and a face mask. Ok that was uncalled for. What dies your wife say?
Guys and gals, we all mean well but I think we need to be a little careful what we say. All he said was he needs tires anyway and this was interpreted as don’t leave town or your life will be in danger. A simple leak from a valve core and a simple fix got blown into four new tires and four tpms valves. I’ve probably put in three valve cores in the last year or so which is why I just stock them. I’ve been putting them in since I was ten along with tubes on my bike. So to a risk averse person with little knowledge, we’ve got him shaking in his boots afraid to leave town. They do lose their seal or get dirt in. I even had a leaky one on a new lawn mower tube. Not unusual. But of course soap the tire to make sure there are no punctures, tire not bald or cracked etc. but geese.
I have several of these valve stem tools that I’ve collected over the years and I also have 5-vehicles with 22-tires between them and I cannot even the last decade that I had an issue with a valve core leaking… It’s just not a thing that comes up. I do keep valve stem covers on all my vales stems so there is no issue of dirt getting in and since I use my own compressor to fill the tires, I do not have to worry about some nasty, dirty tire inflation nozzle left on the ground at a gas station blasting dirt into the valve core… And I use spit to check for leaks… 
He just needs tires and tpms service kits . . . he doesn’t need the actual tpms sensors
The service kit contains a seal, washer, nut, valve core and cap
tpms sensors often last over 10 years, so he’s not due yet, imo
If a tire shop found the valve core to be leaking, they would replace the valve core. It easily unscrews, and a new valve core screws in, takes 10 seconds. If you don’t trust one new valve core, can you trust the other three?
Did you examine this tire for punctures? While inspecting customers tires, I find 2 to 3 punctures each week, I might find one leaking valve core each year. A leaking valve core is an unusual failure, are you sure there isn’t another leak?
People need to err on the side of caution and replace the tires, this way I can use your old tires for 2 to 3 years on my car.
lol … I’ve never had to fiddle with the valve cores on my car tires (other than to remove them for a mount/dismount job), but for some reason my bicycle tires need this part rejuvenated from time to time. There’s a sort of black rubber seal that sometimes gets damaged somehow. I expect the caution here was to dissuade the OP from replacing this part themselves, just before a long out of town trip.
My tires include lawn mower, snow blower, pressure washer, wheel barrow etc. that take more abuse than cars. I don’t know how many I have replaced but I only have a couple left in my parts drawer. It’s not like I buy a dozen at a time. I do remember having to make a special trip at the cabin to replace one for a tire with a new tube. I just find it peculiar that a grown up would consider it too dangerous.
You have to be careful installing new bolt in sensors, to tight and you damage the seal and they leak and have to be resealed, to loose and they leak…
All FSCA’s (as I’m sure others do also) use a kit similar to this, it has a mini torque wrench (you will get the feel for it) a well as a valve core torque tool (black handle)… But again, it doesn’t take long to get the feel for doing it…
Of course it’s dangerous for the first time valve core replacer. 
They loosen the core, they unscrew it and it shoots out with the force of maybe 35 PSI (as a mother might say, “Put your eye out…”), even then, if it misses their eye, they are scrambling around on the ground searching for the missing core (don’t know why…), but if they were lucky enough not to lose an eye and still managed to catch the core in their fingers, they then have to avoid that blast of air shooting out of the tire (it contains over 75% nitrogen and barely 21% oxygen) and they have to be ever so careful not to pass out from lack of oxygen. 
If they had the foresight to position the car so the valve stem is not pointed at some vulnerable target, they might have positioned it straight at the ground, then that blast of air is directed at the dirty, dusty ground and that blast is picking up all the particulate matter and they will have to race and get an N97 face mask to protect their breathing… 
And if they had managed to get the core out, catch it in their hand, and avoid that “poisonous” blast of air, they probably will have to quickly run to the back of the car, and get the new cores, that are still in the package, still in the bag, that they bought at the parts store… 
Well, we know now, the tire has long since gone flat, they manage to get the core screwed in, they then look at the tire pressure machine and see it takesw 4-quarters and they reach in their pocket and the only change it the 37-cents change from the parts store, but as luck would have it, the machine also takes dollar bills… 
They open their wallet and find nothing smaller than a five-dollar bill. But as luck would have it, the quicky-mart is still open. They go inside and ask for change, the cashier says he can only open the cash register with a purchase, so he buys a diet soda for $1.99 plus tax and gets $2.86 back, (surely the two dollar bills will be enough to fill the tire…). 
The cashier asks him what he needed to change for and he says the tire pressure machine… And the cashier laughs and says that machine is busted. He stammers, why isn’t there a sign on it then. The cashier says the wind keeps blowing it off, so I took the hose off, and the cashier asks, did he not notice that there was no hose on the machine? He had not…
The first time valve core replacer is dirty, he’s sore, he’s tired, he’s frustrated, his tire is flat… So what is a first time valve core replacer to do? He drives home with a flat tire… 
So you ask is it dangerous for a first time valve core replacer, the answer is a definite YES and now they are driving home on a flat tire that is slowly disintegrating… 
Yeah quite a scenario that I never thought of. I change my recommendation. See that neighbor riding around on his lawn mower? Just go over and ask for his/her help. Gotta figure if they got the thing started, they are good mechanically. Or if all else fails just yell “MOM” for help.
I have to say maybe the reason I replace more than a professional would is because if one leaks, I just put a new one in. I don’t bother tightening it or cleaning it, just throw it away. Wasteful I know but someone fed their family making them.
Yeah, I can see how that could happen to a valve-core newbie … lol … Best to start the process by letting the air out of the tire first. I try to avoid breathing stale tire air, so I use an old bicycle pump’s discarded hose, open on one end. I clamp the working end onto the valve stem, then all the air leaks out without needing to be nearby.
And it’s always a good idea whenever working w/pressurized air to wear eye protection
Since a new valve core is to be installed you don’t need the old one, let it go. In the time it takes to replace a valve core, 5 psi will be lost.