From your description, it sounds like you’re interacting with a browser. The problem you’re experiencing comes from the browser’s behavior when there is no active window session. Under those circumstances, the desktop disappears and the browser no longer renders the controls on a page. Your click and type into activities timeout becase they can’t find anything to type into.
Now you could remain logged into your robot host machine and never lock your screen and don’t let the screen saver kick in. This would keep the desktop alive but it’s terribly inconvenient and a security risk. Your thoughts of using an RDP session to the robot host is on the right track, but it still suffers from the original problem. When the RDP client desktop goes away, the same thing happens to the RDP session window that happens with the browser. Nothing is rendered in the RDP window so the robot host browser is no longer rendered and you’re back where you started. However there is something you can do about RDP.
There’s a registry setting on the RDP client you can use to tell Remote Desktop to keep the session window alive even when minimized or the desktop is locked. The setting is RemoteDesktop_SupressWhenMinimized that you want to set to the value 2. You make this change on the RDP client and not on the robot host. When you have set this, your RDP client keeps the window alive when connected to the robot host. The robot now renders its browser and everything works.
There’s still one remaining issue. Opening that RDP session isn’t always convenient. There’s one more trick I’ve heard of but haven’t yet tried. You can use a second login session on the robot host with a different account and use that to RDP to the robot session.
Here’s a registry file that will set the value for you.
Windows Registry Editor Version 5.00
[HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Terminal Server Client]
[HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\Microsoft\Terminal Server Client]