Are you using in Unattended Access mode or simple PIN connection ?

The remote machine was using Unattended Access. The local machine used simple PIN connection.

I just had a chance to try it with a third computer. This one is remote (not local LAN). It was using just a PIN. The symptom was that I could log in, and then my view of the remote computer froze. I could see the desktop, but the Remote Desktop window was showing. This window normally disappears when you remote in and a connection is made. It is replaced with the small RD toolbar. In this case, the toolbar was not evident. Also, the person at the remote computer said I was moving things around, but nothing was updating on my screen.
Based on this observation, I can now say that the symptom in all three cases is identical. What happens is that I am able to log in, and then the local view of the screen locks up. My mouse and keyboard are actually working on the remote system, but my view of the remote system is not updated.

This is really strange, correct me if I am wrong but in Unattended Access, everything is working correctly and on PIN access it is not working? Unattended Access and PIN access are the modes for the host, the client is simply the client.

No. It isn't working in either case. In both cases, the screen on my end freezes up, even though they claim on the other end that I am mousing and hitting keys.

And today I saw a new problem: On the new computer, where RD isn't working, I tried logging in to a client's system, and it refused. Said the PIN was bad. But I used the same PIN on my old computer and it worked fine!

Can you verify you're using the same version, and the latest version of the product, on all computers involved?

Yes. And I've already uninstalled and reinstalled it on the new computer.

Another clue: I installed VirtualBox and then installed RD on a VM, all on the new laptop that doesn't work with VM. From inside the VM, RD does work! I think this means that there are no hardware issues, but something in the main OS on the new laptop or in the way RD was installed is interfering.

What security software are you running?

AVG. Another odd thing about this brand new laptop is that it won't run MS Windows Defender, so I installed AVG.

The only way I could find to "turn off" AVG is to enable passive mode, which halts all their real-time protection. I did that, rebooted, and the problem remains. I also went through all the RD menus. Tried different graphics settings, color depths, speeds, etc. No luck.