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tl;dr For normal "desktop computers" USB controllers and related hardware are not capable of capturing video (no hardware to decode DP input into something that computer can interpret) or emulating USB devices ( https://unix.stackexchange.com/a/283147, shows exceptions like some USB OTG capable Raspberry Pi models). But if you absolutely cannot use "remote desktop" or semi-hardware KVM (https://haim.dev/posts/2020-07-28-dual-monitor-kvm/ https://github.com/haimgel/display-switch) and similar approaches - you could add a cheap HDMI capture device to get video signal from the laptop to desktop and some hardware device that emulates USB mouse/keyboard.
Also, controllers that are usually used for USB on the desktop computer side are not capable to emulate USB devices, but there are specialized devices for that (random web search results - https://www.hagstromelectronics.com/usbtousb-shp.html, http://goodfet.sourceforge.net/hardware/facedancer20/, I would also dig into Raspberry Pi with USB OTG + software https://raspberrypi.stackexchange.com/questions/82850/use-raspberry-pi-as-keyboard-mouse-output-to-control-server since it's cheap and easily available, there are probably better options). There are exceptions - e.g., Raspberry Pi Zero is also a "computer" but USB OTG capable https://github.com/milador/RaspberryPi-Joystick https://unix.stackexchange.com/a/283147.