Apple TV app on iPhone won't connect to Apple TV

We have an Apple TV HD model A1625 from 2016. The physical remote stopped working years ago, and we had some initial success with the remote widget on our iPhones. For the past two years though, it's been intermittent—some days it just pinwheels forever and the app "can't find Apple TV". We use two Wifi networks at home, and the TV knows both of them, so we thought that was the problem, but now we can't find it on either. We're running iOS 15.6.1 on iPhone 12 mini. When the Apple TV does work—once every week or so—it works fine, so I don't want to buy a new one. What can we do to "re-pair" or get these devices to recognize each other?