OK that is odd. I'll assume you've already done this but restart both your S22 and Kia headphones, just to be sure you're working on a 'clean slate'. (If those Kia headphones don't have an actual power switch to toggle on/off, do a reset.) On your S22 Ultra, delete any previous Bluetooth pairings to those Kia headphones, then toggle Bluetooth Off, wait a few seconds, then On again. When you try to pair up the two devices, are the Kia headphones still not showing at all?
This is a long shot but try enabling this Bluetooth Gabledorche option. Is your phone running Android 12 or 13?
It's partially functionally in 12 but should be the default in 13. It's buried in the Developer Options menu so there's a two-pronged process to control it. First you need to make the Developer Options menu visible, then you need to scroll though it to get to the Bluetooth options section. (... also there's a chance that Samsung opted to just remove the Gabledorche enable/disable option so it's not showing on your model.) Anyway, the Developer Options menu is an integral part of Android itself, it's just not shown by default so it's up to the user if they need to show it or leave it hidden. Or in other words, it's always there no matter if its menu is or isn't visible. Just be careful not to mess with any of these particular menu options unless you know specifically what's involved, but don't be intimidated by its title or anything like that.
Enable Developer Options
https://developer.android.com/studio/debug/dev-options
Bluetooth Gabledorche explanation
https://www.xda-developers.com/android-13-gabeldorsche-bluetooth-stack/