It's not weird at all it makes perfect sense, the DS4 works with the games that were designed to use it, how is that weird? You can still use the PS5 controller for PS4 games it's the PS4 controller that is lacking the extra functions that keep it from being compatible, if a dev wants to add gameplay features that are tied to the controller they need to be sure the player is using a controller that can take advantage of those things. As far as arcade sticks those are only used for fighting games, there isn't a whole lot you can do with the added dual sense features for a fighting game.
As far as mentioning xbox I also mentioned PlayStation in the same sentence so you would have some perspective on how this has always done, it isn't something new and that the only reason both companies aren't doing that same thing this time is because MS's controller is basically unchanged on the hardware side other than adding a share button. Why you'd choose to ignore the context and just pull out half of the sentence to comment on is a bit strange and seems like you are trying to deflect instead of having a real discussion.
For starters, there is no technical reason an arcade stick can't be used for damn near any game, I know you're aware of that, so I'm not sure why you said "only for fighting games". That's not true, you can use them for any game; as far as I know there are no games that won't let you play them with an arcade stick and as anyone who has ever been to an arcade can attest, Fighting Games are not the only genre that use arcade sticks. Hell, I know I will die a lot but I can go play Dark Souls with my arcade stick if I want, then play Cuphead, Ori, Raiden V, Brawlhalla etc etc, you get the point.
Even if we focus on your exception, Fighting Games are still PS5 games, right? How does all that 'compatibility, extra functions and added gameplay features' not apply there all of a sudden?
Flight Sticks will work too, so will those games somehow be lesser PS5 games?
Again, that's a weird angle to take.
I see Shawn mentioned GT Sport, lets talk about it.
Suddenly the PS4 controller which works perfectly fine with GT Sport on PS4 is a lesser experience on PS5 (I've played that game, might be the best controlling racer with a controller)? Ok, lets accept that as a fact, when you hit rumble strips on the road in GT Sport on PS5 using a PS4 controller it suddenly feels bad.
...but PS4 Steering Wheels are perfectly compatible?
They receive the same exact feedback, and those steering wheels are a more pronounced vibration than the PS4 controller can muster... wouldn't it be exacerbated there?
What if I turn off the rumble, will I be missing out on something? I'm pretty sure that new forcefeedback adaptive trigger thing is going to be optional as well considering that is a pretty big accessibility restriction; at that point, but even then how are all of those other peripherals acceptable but a DS4 is not?
Anyways, I'm not asking you to agree, I know you'll die on any hill unswayed, but you're not convincing me the support isn't a bit weird considering all of the above.