Normally I would definitely agree with you re: noting the condition of the lines being bad. In their defense though, the trailblazer’s undercarriage design DEFINITELY prevented them from seeing this. They had to take off the engine cover that is located in-between my wheels in order to even see where the leak was coming from. None of the metal lines from the radiator were visible until that engine cover was taken off, so the corrosion wasn’t something that was seen during the flush. IF it wasn’t dripping after the flush, as they claim, then taking off the engine cover wouldn’t have been a routine thing for them to have done, as the radiator, on my car, is located on the very front of the car, right behind the grill.
You would be far better-off if you utilized an independent shop for maintenance and repair.
Independent mechanics make mistakes reattaching transmission lines too. I have seen it on a simple radiator replacement.
The important thing is to find a trustworthy mechanic, whether the mechanic works at a chain, dealership, or independently owns the business. Sometimes, if you are lucky, you can find a trustworthy mechanic at any one of these locations.
Hopefully there isn’t any lasting damage, but I will definitely keep you posted if anything occurs. That way there is a record of the story from beginning to end, should someone else run into a similar problem and go looking on the Internet for guidance. 
Transman is correct and this facility has fed you some BS.
Your transmission IS damaged, no doubt about it. The power loss over 10 miles and the vehicle’s failure to pull itself into the driveway is proof of that.
A pan drop and filter change is always done before a flush (if done properly) and these guys are feeding you a line of bunk about the reason for not changing the filter.
Any “gunk” as they refer to it will almost all be in the pan, not randomly scattered throughout the transmission. With the pan drop the gunk will be cleaned out and there will be very little left anywhere for the new filter to catch.
Keep this in mind also. If the pan has a lot of “gunk” in it then the transmission is likely doomed anyway because that gunk is often friction material caused by slipping.
The reason for performing a flush only? It’s speedy and profitable whereas a pan drop/clean and filter change is much more time consuming and messy.
A facility can perform half a dozen flushes in the time it takes to perform one pan drop and flush. Which do you think they would prefer doing? It’s all about the $$$.
My earlier point was that if someone got a bit rough (in a hurry maybe?) while disconnecting the line this could have caused a weak line to become damaged.
Flush pressure did not cause this because the flush machine uses the pressure normally created by the transmission itself. Something to keep in mind and best of luck.
Thank you for the clarification re: the drop pan and filter change. Even though they tried to “make things right,” in light of all the great information I have received from you all, I think I will err on the side of caution and make sure I use a different garage from now on for my oil changes. I will keep in mind the information re: the tranmission damage and hope for the best, I guess. Is there anything I can keep an eye on to know if/when transmission is going bad? Or should I take it to a transmission garage now to have it assessed/fixed?
Having read the posts and the follow up “results”; at this point I would recommend taking some further action to preserve the transmission. That action would be to have the trans pan dropped, cleaned, and a new filter installed. Then make sure the proper fluid is put in the trans. The best place you can be sure to get this done with the proper fluid would be a GM dealer. A good independent trans shop is fine, so long as they are sure to refill it with the correct spec’d fluid.
After the flush was done and lines replaced I still don’t read a new filter got installed, and it should be done. Many chain shops use “generic” fluids that they state are good, but these fluids don’t always perform properly. There are getting to be so many different auto trans fluids that a chain shop won’t want to stock all of them so they use “generic” stuff, not good.
I’d say have a simple pan drop/cleaning and change the filter followed by topping off the fluid.
What you really want to know while the pan is off is what’s in the bottom of the pan.
A very light, blackish looking film is (comparatively) not too much to worry about. If the person that pulls the pan can drag a fingertip over the bottom of the pan and end up with a glob of black or dark brown goop on their finger then the transmission is in trouble.
If the worst case scenario is true then just drive it until the trans dies and cross that bridge when you come to it. Any failure will likely not be sudden so you’ll have plenty of warning but I would not take any long out of state trips in it.