tv remote lost

Yes. If its the silver infrared remotes, they do not need to be paired. and can be used directly between different Apple TVs.

If its the black Siri / Apple remotes, then you can use an iPhone or iPad's with the remote widget instead: Set up the Apple TV Remote on your iPhone, iPad, or iPod touch - Apple Support

Otherwise, yes, you would need to constantly pair and unpair the remotes with the different Apple TVs.

Hello,

You would then need to purchase a new Apple Remote.

Hello,

While it is possible to pair and unpair Apple TV’s remote, it is a huge pain and not recommended.

If your Apple TV remote isn’t working - Apple Support