Personally, I wouldn't use this to switch between three screens at the same time, but rather using it to not have to pair it every single time I use the keyboard on a different device. That's quite frustrating.

That would be also my plan, but once I have this setup, it would be impossible for me not to constantly switch between devices.

Pairing every single time is actually a plus for me: It acts as a barrier to avoid multitasking in different devices.