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.
https://github.com/debauchee/barrier
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.
https://www.eklhd.com/product/kvm_41dh.html