What should I do about this ongoing tire problem?

Did you check the tire pressure with a regular tire gauge? If that shows OK, maybe you have a bad TPMS sensor or the sensor needs a new battery.
Ask around in your own area for recommendations for a good tire shop.

 

Or you have a bead leak where the tire doesn't seal properly with the rim of the wheel. Definitely take it to a good tire shop, or some place like Costco that has good policies for the buyer.

 

Was a new tire really needed, i.e. was the old tire not patchable? The center is usually repairable.

If they didn't give a credible explanation for why the original tire should not be patched, but simply moved to sell you a new one, then they were somehow taking advantage of you. I'd never return to them.

If the stem itself was loose, on only the tire they worked on, then that employee is the first suspect to consider as the "someone" who is messing with you.

Three inflation problems and your own pressure gauge still hasn't come out? If it is lost or buried or inaccessible, then go buy a new one. Now. Buying a new one will be cheaper than repeated trips back to tire shops for troubleshooting.

Any respectable tire shop ought to be able to fix this. Since you clearly have lost confidence in this chain/location, then pick another shop or chain or location. But I'm too far away to have any clue of the tire shop choices in your area.

 

What Fuzzy said. You should only need to replace the tire if the screw or other object is embedded in the sidewall, not the main tread.

 

OR.....if the object in question left an irregular, jagged hole.
BUT the default for most ALL tire shops is to sell you a new tire whether you NEED one or not.
I'm surprised that they didn't INSIST that the other 3 were unsafe too.

My best tire experiences have been at Sams Club.....for years.
Recently had a puncture plugged at WalMart too, just because they are close.

Just go to another tire shop to get the right pressure in all of them AND inspect the caps after to be sure all are tight.

And don't totally discount the possibility that someone IS messing with you.

 

Replacement will also be needed if one of the belts or cords is cut. But most of the time this doesn't happen. Shops have been willing to patch most of the flats I've taken to them.

 

Do I understand the OP's issue?
Wants a local tire shop they can trust?

Making sure the valve stem caps are on, checking tire pressure when the tires are cold, setting the preferred tire pressure and resetting the TPMS dash setting (owners manual show how), tightening the valve stem TPMS attaching nut (noting a possible o-ring) all seem to be at home do it yourself items.
Home Depot sells a nice 12 volt tire inflator (Husky brand name) for about $35.

FWIW In my area a good tire shop is somewhat hard to find. So you are not alone in your quest for a good shop.
Again not sure if I understand what type of help, comments, witty remark , the OP is seeking.

 

YES.
And that's impossible to answer unless you live in the same area........since competency can vary WIDELY from one brand name shop to the next within the same brand.

 

Your original tires (at least the three original that are remaining) are certainly ready for replacement. That should fix your concern with having four reliable tires.

I've always been happy with Discount Tire. They match (and sometimes beat) internet prices. They offer good service: free pressure check, free rotation, prorated warranty if tire is damaged.

 

At 6 and 7/32nds remaining, it seems rather early to replace them.

 

Holes from large screws can still be plugged. I picked up a #12 wood screw in front of a construction site once and my local tire place was more than happy to patch it.

I'm surprised the shop only sold you one tire; typically the recommendation is to replace tires in pairs so you don't have slightly different diameters and different friction coefficients/grip by having the same tire model with the same tread depth on both sides.