The official Tesla tire repair kit is made by Fix & Go, who also make OEM kits for many other brands like Jeep, Dodge, Chrysler, etc. If you scroll down on their website you will see a big list, and some of their inflators for other brands look just like the Tesla one
Tire Repair Kit for Effective Flat Tire Repair | Fix&Go
I've had the Tesla-branded inflator for more than 4 years and I didn't have issues with the 12v plug even after extended sessions airing up trailer tires that need to be at 95 or 105 PSI. Where I did start having issues after a handful of uses is in the same place you are -- with it shutting off by itself or not starting up unless you mess with the selector knob (or more reliably for me, put pressure for down on the area above the power button). The problem is that it uses a very tiny relay to turn the pump motor on and off when you press the power button. That tiny relay is not up to the task and sometimes you need to press down on it by pressing down on the area right beside the power button in order to get the pump working. I ended up replacing the circuit board and tiny relay in my inflator with a larger 30A relay.
I used up the flat repair liquid port early on with a coworker's flat tire. This was before Tesla offered replacement cartridges of the repair liquid. So I contacted both Tesla and Fix & Go to offer a replacement and eventually they did. The second cartridge of repair compound was used on an ATV tire, and I didn't buy anymore. So I never got to test it out on the OEM tires with foam. However, I did use a standard tire plug on one of my tires with foam just like I would on a regular tire and it lasted the rest of the tire's life.