MacBook Pro and Mac mini sharing Studio display. How to do it?

I have a MacBook Pro and a Mac Mini that I want to use with a single apple studio display with the ability to switch from one device to the other without pulling the thunderbolt cable in and out each time. I thought this must be simple but it seems not. The display only has one input (why?). I've looked at third party KVM switches, I'm not sure which would work at all, which would give a full 5K resolution and power supply to laptop. And some of them seem crazy expensive.

My second problem is that it's very awkward trying to switch the pairing of the magic keyboard and mouse between the two devices and it always seems to involve having to plug them in by USB and connect that way.

Am I missing something? Is there really not a streamlined way to do this? Surely loads of Mac users must be in a similar position?