My AirPods don't show anything on the iPhone screen after initial pairing for me (iPhone 6S with iOS 13.0) but work fine. Not long ago one Pod did a weird ball bouncing sound effect and wouldn't work and the case still had enough charge and the Pods showed a green light while in the case. Never found out what that sound meant and my Watch saw the Pods but refused to connect (spinning circle of eternity). I ended up rebooting both Phone and Watch and they worked fine afterwards. Never figured out the issue but my iPhone wouldn't do anything with BlueTooth and my watch was on Wifi despite my phone being next to it. Must have been a driver crash on the phone causing the whole thing. Try rebooting the devices you use your AirPods with.