I just spoke with a Senior Advisor at Apple about this issue.
My setup: wired mechanical keyboards (I have about a dozen) connected to my iogear GUS432 KVM connected to my LG 38WK95C display's USB hub connected via USB-C to my 14" M1 Pro MacBook Pro.
My flavor of this issue: the other 3 peripherals connected to my KVM - mouse, mic and webcam - all work fine. It's only my keyboards that don't when connected via the KVM. Also, my keyboards work normally when connected via KVM to the other machine on the KVM - a Windows-based Intel NUC. Finally, the few keyboards I have that can also connect wirelessly via Bluetooth work fine over Bluetooth, just not in wired mode via the KVM.
We ruled out the LG display's USB hub by connecting the KVM directly to the MBP so that the connectivity is like-for-like with the Windows machine. Still no dice.
The SA said that most likely there was a security vulnerability that was addressed in 14.4 and that in such cases where third-party peripherals are at play Apple typically won't send these cases up to Engineering. My options are to either downgrade the machine, which requires a complete reinstall, or to reach out to iogear as manufacturers will need to conduct their own testing and if they find this to be a big enough problem, they have more clout with Apple than you or I as individual customers.
This truly sucks. I've been happily using this KVM for 2 years.