no mate just copy the game over or hit the launcher from your external drive, it should put whats needed back on your ssd. well it did for me last week anyhow.
That's a bit surprising - but then again we don't know which OS either of you are talking about.
On Windoze that would indeed be a rare occurrence as most programs install critical registry entries that are required to run properly. There are exceptions, but they are rare.
In addition, I think Elite requires some C runtime files that might need to be installed on a new OS refresh. that's probably what the OP doesn't have on the new OS install. There might also be some DX9 files needed. I didn't pay close attention as my installs went very smoothly, but I do recall seeing some older runtime components being installed that are needed for Elite to run properly.
Might even be some older Asp.net version re-install that doesn't get automatically get installed on new OS versions. 3.5 I think.
Mac systems may not have the same issues.