One summer day in 2019, I was dismayed to find I had driven over a nail or screw. I didn't even realize - my Mazda's TPMS flashed the yellow triangle on my dashboard, indicating low pressure in one of my tires. Luckily, my tire wasn't completely flat. Upon further inspection, I found the culprit. Furthermore, the screw was smack near the center of the tread, which is actually a good thing in terms of being a good candidate to be able to patch it.
You may have read or heard that Discount Tire (known as America's Tire in California) offers free tire repair / patching. I can confirm first-hand that it's true. I didn't even buy my tires from them. They were busy too, being Labor Day weekend in 2019. They even apologized for taking longer than usual! I felt bad for taking up their time and paying $0, and this is the reason why Discount Tire can afford to do this. It's just brilliant marketing, customer service, and a bit of psychology. They didn't try to sell me on anything. I do get some occasional e-mails on new promotions, but I could unsubscribe anytime. The next time I need to get new tires, I will definitely remember Discount Tire, and may feel like I need to "pay them back". But for now, my car is back on the road, and my wallet is no worse for the wear.
Just know that they won't be able to perform miracles. The puncture must be away from the sidewall (see diagram above). If your tire is flat, DO NOT keep driving on it. You will damage the sidewall such that it won't be safe to use, even if the original puncture is patchable.
Check out my invoice to see proof of the free tire patch at America's Tire: