I have the ATV 4K 2017, not the HD model, and my iPad Air 2020 is running iPadOS 14.8. When I select Search nothing happens until I cursor down with the remote to the letters and numbers. Once I cursor into the letters and numbers I get the notification on my iPad Air 2020 that I can use it for the Remote. Tapping the notification displays the Remote on my iPad Air 2020 and then I use it to enter text strings. So with my ATV 4K 2017 and iPadOS 14.8 it does still work. I don't have iPadOS 15 to test, nor do I intend to install it at this time.
OP, which iPadOS is your iPad running. Perhaps Apple changed something in iPadOS 15 if you are running it. I know they screwed up a lot of stuff in 15, and that is why I downgraded back to 14.8.
To put the Remote app in the Control Center, on your iPad go to Settings --> Control Center --> select Apple TV Remote.