The PS3 can use the DS4, but only wired. Can it use the dual sense as well? (I haven't tried)
The only reason I can think of why DS4 is not supported on ps5 is that Sony wants the new features to be used and not have developers test / tune their games to work with DS4 as well. Of course with cross-gen games it doesn't matter, yet going forwards they likely don't want to have to keep supporting DS4 on PS5 for the while generation. It's just a bit of extra QA but also a discouragement to do things that won't be possible to play with DS4.
Ratchet and Clank and Astro's playroom aren't the same on DS4, some things in Astro's playroom not even possible with DS4. Having to add workarounds to support DS4 is extra work.
Yet true, why not simply allow it for ps4 games and cross-gen games. On an at your own 'risk' basis. Disclaimer for games, best played with Duelsense. Perhaps the wireless protocol is different on ps5 adding some hoops. It could still work wired. Maybe it will come later, Sony is lagging behind in QoL features on ps5. The UI needs plenty work, features are still missing (VRR, 8K) and still needs plenty stability updates. (Yesterday I kept being interrupted with a periodic "Something went wrong" WS-115506-08 error, press OK and continue. No clue what it was, but kept popping up every 10-15 minutes)
Btw, you can also share play to a ps4 and use a DS4 on that. So if you still have your old ps4, share play to there, then toggle the tv input back to ps5 and play with DS4 that way.