You could also try using KDE Plasma instead of Gnome, which survives GPU resets.

Okay, then the driver doesn’t allow Plasma to turn it on for some reason. You can report that at https://gitlab.freedesktop.org/drm/i915/kernel/-/issues, maybe someone from Intel can help.

No I can’t move my mouse to the external monitor.

Do you mean that it stays on the internal display, or that it doesn’t show up on the external one?

Or to ask it differently, can you move a window where the monitor should be?

My laptop detects the monitor correctly so it shows up in the KDE plasma display configuration but the monitor is not getting any input.

So the system thinks it’s enabled? Can you move the mouse cursor to it?