I think the reason why you won't see any big exclusives on PC is that because consoles are close to PCs, big games should be ported to console otherwise you're missing out on huge markets.
Beyond the live service type games I mentioned earlier, there's also a ton of smaller indie-AA type titles that are PC exclusive. Even that is dwindling down though since porting gets easier and easier for indie devs
Yeah, but today it is mostly the other way round: Big console games get ported to PC. Exceptions like Cyberpunk 2077 (which was a PC-first development) are increasingly rare. Smaller games are another matter.
Logging in to the Linux side isn't even part of normal use for a PS2 TOOL. The Linux system runs the RPC so you can connect it as a target in ProDG or whatever, then it pokes your code into the memory of the PS2 side.