Does the remote keyboard not work if the Apple TV is connected with Ethernet?

Are you referring to the on-screen keyboard of an iOS device, for which you may be prompted when text input is requested?

That relies on having a matching account for iCloud, as well as Bluetooth, and the same network.

An ethernet connected Apple TV is fine, if the router bridges the wired and wireless into one logical network.

Enter text on your Apple TV - Apple Support

• Use a nearby iOS device to enter text on your Apple TV

Welcome to the Apple community.

Whilst none of my Apple TV's are on ethernet any longer, when they were I had no issues typing text from the remote app.

Whilst the Apple TV is connected to ethernet, you will still need wifi available (same network) for the remote app to connect.