I went back and forth with SysAid several times trying to get them to provide proper uninstallation methods.
In the end, I had to figure it out by myself.
In case anyone else out there needs a reliable way to nuke installed versions of SysAid so you can properly install a new client, you can run the following powershell script from an admin ps console. The only thing I didn't get around to was code to find and remove start menu entries. (you can find it in the code by #stub and add it yourself, if you'd like)
I'm also still not 100% sure this gets everything, but it removes enough to allow a reinstall in cases where SysAid refuses to run properly. While this did work on over 150 systems in my environment without issue, I do not guarantee this won't break your computers since it is deleting registry keys, though I did add some logic to avoid that. Try it on something controlled before deploying to a large group of computers.