Apple obviously doesn’t want it because even under the best of circumstances it’s difficult to ensure consistency from repair to repair, much less between different technicians or locations. And explaining a customer that while, yes, they did go to an authorized repair shop, but no, that specific repair doesn’t fall under Apple’s warranty or seal of approval, is also a nightmare.
Similarly, for keeping parts on stock I can see both sides of the debate. Shops would like to have stuff in stock to provide the best and fastest service available, Apple on the other hand wants to prevent parts from falling in the wrong hands given the already prolific grey market for their parts.
I know they’ve already loosened up a bit with some shops that have high volume and good track record, letting them have some common stuff on hand (e.g. batteries, screens, etc.), but there’s also the option to let the customer get the part via the self-repair program and bring it to the store to have it placed.
There are two competing interests at play but I think with a bit of compromise in either end a workable solution is possible.
As for shops getting pissed when asked if they’re part of the program, I think your explanation is rather poor. It’s equally, if not more of, a bad look to get pissed everything they’re asked that question. If anything it makes them look sketchy when that doesn’t need to be the outcome.
People are generally just interested in the quality of the components used and want to know if official components are used to gauge that, as well as know if they can go to Apple should problems arise (i.e. if it’ll be covered by Apple’s repair warranty).