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IIRC, the USB-C connector on the RPi4 is connected via USB 2.0 port and only the two blue USB ports are plumbed to as USB3.0.
There are two methods to get video over USB. The first, older method, is to present a video card to the host as a USB device. Image rendering is performed by the connected device. If there is an ARM driver available, these should work with the Pi.
The second, newer method, is to use a USB-C 3.1 alternate data mode. This allows host-to-device communication over some of the additional pins. This also requires some internal plumbing between the GPU and the USB port, which the Pi lacks.
I had a similar need to share monitors with a Pi, personal desktop, and a work laptop. I ultimately bought a CKL HDMI kvm and it has been working really well. Since the Pi only supports dual displays, you could switch two monitors and leave one permanently on the dock.
https://www.amazon.com/dp/B07KTMVKWQ/ref=cm_sw_r_cp_apip_gee8hsvOcGa3P
