Am using 1.18.2, big modpack, but also tested without any mods other than cc:tweaked and this one, when I use the keyboard left clicking the screen works, but right clicking and typing don't.

was using advanced monitor btw.

Update: used part of your example code, typing works, but right click and scroll wheel functionality is missing, I'm not sure if any of this is possible to remedy, but I'm especially unsure of the possibility of this one: drag clicking also doesn't work. No pressure if you can't do some or all of this.

You have to use my monitors with the GPU block for full mouse support.

The advanced monitor only detects a right clicking on the monitor in world. The keyboard just simulates right clicking while the GUI is open.

Is there a way to make a gpu monitor run a program like you can on an advanced monitor?

You have to write a cc terminal emulator, that can display on the GPU monitors.

Do you know by any chance if one already exists or if it would be hard to code?

I uploaded a terminal emulator to the examples:
https://github.com/tom5454/Toms-Peripherals/blob/api/examples/termEmuTest.lua