"Apple TV Keyboard" notification not appearing

My girlfriend and I both have iPhones.

Both our iPhones have been paired with our shared Apple TV using these steps:

https://support.apple.com/appletv/pairing

We've both signed in with our Apple ID accounts on the Apple TV using these steps:

https://support.apple.com/en-gb/HT204053

However, only my phone receives notifications to input text with these steps:

https://support.apple.com/en-gb/HT200107#nearby

Her iPhone never receives these notifications.

I have checked her iPhone settings to see if notifications are disabled (Settings > Notifications > Apple TV Remote Keyboard). However the "Apple TV Keyboard" does not appear on her phone (but it does on mine).

I have tried putting my phone in airplane mode (turning off Bluetooth and Wi-Fi), but even with that her iPhone still doesn't receive these notification.