AFAICT, Apple excludes all their own apps. Seems like a way to try to get people to use Apple's apps over everyone else's. Just to check, I pick the Quick Time Player. I pick "New Screen Recording". No prompt.
But that's also you performing a positive action to start the (presumably time-limited) recording which is different from something in the background passively recording your screen over a long period (Bartender, etc.)
(I do not like the permission prompts but I understand where they're coming from)
Don’t let facts get in the way of Apple bashing. Apple already does so many legitimately anticompetitive things (see: iOS EU blah blah), so resorting to this low-quality complaining is very lazy.