The blower on most modern cars runs all the time when the ignition is on on the slowest speed.
I also find it impossible to believe that during this inspection and the ensuing 2 weeks of ownership that not one facet of the mode control was ever used in regards to heating, A/C, outside ventilation, etc.
You’re insinuating the mechanic did not turn on anything at all and feel that he owes you for this failure. No way, no how, never.
Mr. Green has given the cause; Father Time.
You essentially bought someone else’s used pair of shoes. Sometimes they don’t fit quite right and sometimes things come apart on them. Even brand new shoes have been known to fall apart now and then… 
I am going to make a prediction: This vehicle will have a transmission problem sometime in the future - provided something else doesn’t happen to take the vehicle out of service.
I agree with Capri Racer.
Whose fault will it be at that point?
"Who’s Responsible?"
That’s not important. You got more than $350 to $376 worth of car buying education. You got a real bargain !
Here’s what you learned for a measly few hundred bucks:
Stuff happens on 6 - 7 year-old used cars with over 100,000 miles even when checked out by a mechanic.
No car in this price and miles range is perfect. Always plan on some minor repairs/maintenance/tweaking when buying used to save money.
Used cars need to be purchased with complete bumper-to-bumper warranties or you’ve got nobody to blame, but yourself.
You can always buy “new” or “Certified Pre-Owned” and get factory warranty protection. Then the manufacturer is responsible. Of course, you’ll probably pay up-front for their taking on that responsibility.
There are reasons why used cars are cheaper than new cars.
Your experience is “normal.”
Cars are expensive to buy, maintain, repair, operate, drive, insure, fuel-up, replace, etcetera. Try keeping four or five going, simultaneously, like I do.
I buy used cars, seldom new cars. I buy used and plan on tweaking some things and spending a couple of bucks to make the car “perfect.” I buy Certified Pre-Owned and let the manufacturer be responsible for tweaking.
Best of luck. Enjoy the “new” car,
CSA
Now I ask you-have any of you ever purchased a home? Did you get it pre-inspected? Did you rely on the expertise of the inspector? What if after a month you found you had mold in your walls?
Here’s the flaw in your LOGIC…Mold takes weeks or MONTHS to form…It’s NOT going to form over night.
There’s a very good chance the Fan WORKED when the mechanic inspected the vehicle…And then 2 weeks later…It didn’t…Sorry…but you’re still WRONG.
With the bang you describe comes the probability of a foreign object getting in the fan. Do you or your passengers put ‘‘stuff’’ up on top of the dashboard ?
Any of that stuff missing ? ( like pencils, pens, paper clips, hair clips, coins, etc )
With a used car there could also be debris already in there that you knocked in to place. When someone repairs it they’ll get to see what’s in there.
( I stopped putting any and all stuff on top of my dashes when a Bic lighter exploded in there once ! )
The next probability is an age cracked fan working normally on low or medium. When you finally turned it to high…
Me thinks the OP has left the building.
When I first read your post I was expecting my engine needs a $2000 top end job since the timing belt snapped(replacement item a mechanic cannot inspect life of).
However it turned out to be a blower fan thankfully for $350. Like others said it could have broken the day the mechanic checked or when you turned on the heat. I will add mechanics typically check blower operation as a check to make sure AC/defrost works.
Unless I’m mistaken (have been before, will be again), this system, regardless of whether you run A/C or heat, is always on - unless you specifically turn it off, every time you start the vehicle.
Regardless, I think I’m with TSMB…the OP has left the building.
110K is a lot of miles even for a Honda. I don’t think you can expect everything to be like new. There’s many things that start to go wrong at this mileage. I’m not dissing the car. It should be perfectly serviceable for many more miles. With proper scheduled maintenance all the problems you are likley to have are all manageable.
I think you have to expect to have this kind of problem though. I have a 70’s Ford truck that has had a noisy blower motor for its entire life! I just live w/it. The mechanic who did the inspection isn’t to blame in any event. There’s no harm asking the dealer for some help. He may be willing to do it for parts cost only. There’s no harm in asking. Keep on the good side of your mechanic that did the inspection. Sounds like he is a good resource for you to use in the future.