If the blower stops and/or slows it just about has to be the blower fan or it's wiring. Not getting flow out of one of the ducts is usually from obstruction or blend door issue.
I had a 90 jeep cherokee that did the same thing. If I remember right it was the blower fan [mine also did it in heat mode. That was 20 or so yrs ago and my memory is a little fuzzy.
The only times I've ever replaced that washer was when it didn't seal correctly which is rare. I bought my 2010 JK new and it still has the factory drain plug washer.
Why not just buy a set of torx bits?
https://www.autozone.com/screwdrivers/driver-bit-set/p/duralast-3-8in-drive-torx-bit-socket-set-5-piece/565116_0_0?searchText=torx%20bits
IMO it would be best to dig up the title [or even registration] which will have the yr stated on it. Jeeps keep the same style for several yrs so it might be difficult to pinpoint the yr just by the body style.
I buy most of my parts at AutoZone opting for the parts with a lifetime warranty. I've gotten good service out of most of their parts although I can't remember the last time I had to change out a water pump on anything other than my tractor.
What is the engine temp?
If the hose going to the heater core is cold once the engine has reached operating temps there must be some sort of restriction so I'd start by disconnecting/inspecting it.
If the thermostat was faulty you'd either over heat or the engine would take a long time to reach...
It can be used to remove/tighten the valve core along with depressing it to remove some air.
I have one somewhere, don't think I've ever seen the blue one before.
Does it need to be on the exterior? Mine is mounted to the inside of the windshield in the neighborhood of the mirror. I'd have to go look to see what brand it is - a gift from my son.
Powder coats typically dry to a harder film than regular enamels which is why they normally outlast regular paint - it also makes them harder to prep for repaint.