I've got a nice desktop setup at home with dual monitors and a company laptop that I'd like to integrate as seamlessly as possible. The laptop has a Thunderbolt port and I do have a Thunderbolt dock, but hooking that up in addition to a KVM switch would be a nightmare of cables. Are there any switches out there that can accept a Thunderbolt connection for one PC and a standard DisplayPort/USB connection for the other?
I got my wife a USB switch for the camera, keyboard, and mouse, and a separate HDMI switch for monitors. Ya, it's 2 buttons, and 2 cables to plug into the laptop, but it seems to have worked well. Maybe look into that, but Thunderbolt?
I use barrier for sharing one computer's keyboard and mouse with other computers. No issues with it. I have a mac mini as my main computer and use barrier to connect to a MacPro with its own monitor and my laptop on a TB3 dock. Works great after a bit of set up. You need to be on the same network with the right ports and protocols opened so if the work laptop is on a VPN doubtful it would work.
You can go USB-C to standard display port in just a cable. I about to test an EKL-HD 16 port HDMI switch in a day or two but they make lots of switches, extenders etc. Supper cheap though hence the test.
Logitech makes a line of mice and keyboards (MX) that lets you pick from one of 3 bluetooth devices. MSFT has some too (their Surface line of accessories)
Pair this with a USB-C hub and it'll do exactly what you're looking for. If you get a PD capable hub it'll even charge your laptop via USB-C, eliminating 1 cable (on the desktop - still need one running to the hub)
USB-C hubs are a little pricey unfortunately. $50-450, depending on what you need.