Well I'm not sure that screen-scrape is need here all the users need to check is that the app(s) are live and kicking and they can do that just by looking at the UI if it is dead, kill it (Task Man) a restart it no further acction is required. The rest of the information is (verbosely) logged to files.
I tried alternative solutions here, I even managed to scrap half of those apps. The part that actually handles the transport and persistance part for 3rd party systems I rewrote it entirely in C#, it can now safely run as a service, I made it more resilient a various levels, it has been in production for quite some time now as smooth as silk. In fact I earned a gigantic bonus because of that 
However the rest of that suite (POC is a better term), handles the business logic, not to mention that it is supported/maintained by our headquarters in Europe. And this time I had to agree with my boss and rule out a replacement because it's too risky. Unexpected behavior for the rest of the solution, and non-company-standard implementation, is on top of my list.
Anyway, I even tried the scrapping-the-damned-POS approach, and took it as far as I could go. But that still left us struggling with the "really 1 instance has to run 7x24 no matter who is on the box" problem
I don't know if that can be accomplished by the RDP client app you mentioned.