Your PS4 controller not connecting is frustrating. Here are two ways to fix a controller that won't connect wirelessly.
At the time of writing, only the beta Steam client supports DualSense, so you need to switch the Steam client to beta first. Click 'Steam' displayed in the upper left of the Steam client, and click ...
The PS5 controller doesn't just work with the PS4, so you must pick up a Bluetooth adapter. This will usually come in the form of a USB device which needs to be plugged into the PS4. The adapter then ...