I have the “1.4 Display Port KVM Switch - Dual Monitor - Four Computer” KVM and while my Windows computer (1080ti), M1 Macbook, Work computer (Dell Laptop) all will wake from sleep or wake the screen; Debian will not. I am running a AMD Ryzen 5 3700 with an AMD RX 580 4G. Usb activates (mouse and keyboard light up) but the screen wont respond, except by restart. I have tried disable all power saving but still the same issue.
Probably linux settings try using xrandr to lock the desktop config. Ssh in and look for clues in the system log. Nothing in the kvm will really get much in the way here.