It sounds like what you are looking for is some sort of VNC connection which will give you direct access to someone else's machine.
A web search for "
remote computer vnc" will turn up a number of articles that might help.
Unfortunately, I don't have any personal experience with this, at least not for many years. Back in my Windows days, when I was wearing a headset, I was allowed access to a remote computer by a client's IT department, but I didn't set it up, they did.
Now that you've provided more information about what you are looking for, perhaps someone with more first-hand experience than I will join this thread.