You would bring your own mouse, however when you sit down at a tournament machine you will need to install specific mouse drivers and get it configured. Being able to sit down at any machine and just plugin your mouse/kb and have a familiar setup seems like a great idea to me.

You would still need custom drivers to use the cloud mouse.

Why not just give each mouse a unique ID then, sync the settings anonymously against that ID. Don't even have to log in anymore

Correct, pro-gamers (and amateur competitors too) will take their mouse to any tournament, and if they forget it they will try to find the same model to use.

Pro-am gamers bring the rest of their PC hardware to the tournament.