I solved the multi-computer-to-multi-monitor problem with a Level1Techs KVM[1]. The price (~$500 for the variant I purchased) initially gave me pause, but the longer I've had it, the happier I am with my decision.

It handles all the switching at the hardware level and thus has no perceptible lag for video or anything else. I'm able to connect a single set of peripherals, in my case, two monitors, a keyboard, a trackball, and a USB audio interface, to both my Linux desktop and a CalDigit Thunderbolt dock connected to my laptop. The L1T KVM has hotkeys[2] that let me switch between systems, with only a 1–2 second delay.

The benefit, for me, of this extra box now mounted under my desk is that when I upgrade my monitor, I only care about how good a display it is, not whether there's some perfect confluence of KVM, refresh rate, aspect ratio, display technology, etc. I find the monitor I want and let a separate IO routing layer.

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[1]: https://www.store.level1techs.com/products/p/14-kvm-switch-d...

[2]: https://forum.level1techs.com/t/official-l1techs-kvm-faq-ult...