Did anyone here ever try to do the Coop Any% run while controlling the two characters? It's possible in Local if you bind both characters on the keyboard.

I've done it before with double Huntresses. But it's basically impossible to do Bandit Sprint with this method so it'd kinda outdated.

Why is it impossible? Is it because both characters are confined to the same screen?

Yes, and you'll just lose track of the one that doesn't have the movement speed boosted. The main obstacle of solo coop was always to not accidentally get movement speed items. :>