What's missing is you're using an AirTag for something that it's not designed to do ... and that is, for tracking. Whether that is for pets or people.
The reason is the AirTags do not rely on GPS, but on both Bluetooth and the Apple Find My network.
Once the tag is outside the Bluetooth range of your iPhone (~30-40'), it will require another iPhone (that is participating in the Find My network) to pick up the tag's Bluetooth beacon signal. Once one does, it will relay the tag's ID, along with the iPhone's GPS location, to the Apple servers over an Internet connection. This where you get the updates from on the Find My app on your iPhone.
If your goal is to keep track of your cat, you will need to consider getting a dedicated GPS tracker for it.