When I purchased the MX Keys + MX Master 3 combination I was convinced that the Easy-Switch functionality of the MX Keys would switch the connected computer for both the keyboard and the mouse at the same time.
It's major disappoinment that this is only switchin the connected PC for the keyboard. Having to flip the mouse and press the switch button on the bottom of the mouse multiple times is not 'easy' at all.

It should technically be possible to achieve a real 'easy-switch', meaning both devices with one tap on the keyboard. If it's not possible due to BT limitations (i.e. not possible to communicate from MX Keys to MX Master directly while connected to a PC), then it can always be achieved by letting the current connected PC send the switch signal to the still connected mouse, before the MX Keys connects to the selected PC.

E.g.

1. MX Keys and MX Master are connected to PC1.
2. Users presses Easy-Switch 3
3. MX Keys sends signal of incoming 'Easy-Switch 3' to PC1 (Logitech Options)
4. Logitech Options sends the 'Easy-Switch 3' signal to MX Master
5. MX Keys and MX Master switch to Easy-Switch 3 profile

The switch can happen asynchronous, so it's not necassry for the MX Keys to wait until Logitech Options confirms that it was able to inform MX Master of the profile change. It can switch to the selected Easy-Switch as soon as it has communicated the switch back to Logitech Options.