Problem
Playstation controllers (DualSense, DualShock 4, and possibly others) fail to connect via Bluetooth. They immediately disconnect after pairing.
Cause
A bug in bluez version 5.69 introduced this issue. Version 5.70 is still affected. A fix should be present in version 5.71.
(You can see your current version by running rpm -q bluez in a terminal).
Related Issues
Bugzilla report: #2247548
Upstream issue: PS5 controller will no longer pair with bluetoothd · Issue #614 · bluez/bluez · GitHub
Workarounds
Until the fix is released, connect the controller via a USB cable.
Very experienced users might try downgrading to bluez 5.68 or older, or building the latest bluez with the fix included. But this is not recommended to a general user.
You can discuss this issue here.