It just quits when trying to install patch
Not sure what this is - how far do you get in the launch process? I did a one-off synchronisation of Steam & GeforceNow libraries when I first started, now it's:
1. In the GeforceNow client select the Steam launcher in the Elite link and press "Play"
2. It instantiates the virtual machine (I assume that's what's going on, with an Elite-y backdrop) then loads the Steam launcher using saved details (IIRC on the very very first launch on GeforceNow I logged in to this using the same Steam account I used to buy Elite, and had linked to a Frontier account)
3. I press "Play" inside the Steam launcher and the Elite launcher starts
4. The Elite launcher comes up (which tells you if your Steam account is linked to a Frontier account), I press "Log In", then "Play" to actually launch the game. If the game client is out of date 'cos Frontier have just released an update then you see "Update" rather than Play. Update doesn't really work - need to wait for Nvidia to update the game client (has been same-day for Steam in my experience).
In step 3 I am often presented with a black screen, I just escape and press Play again and it works (in fact that's my muscle memory now).
In step 4 maybe 5% of the time the launcher comes up saying my Steam and Frontier accounts are not linked, I just quit and restart the whole thing and it works fine next time.
I've had a quick rummage on Google and can't see a consistent answer to the message you're getting - sometimes it's "just started working" a day or two later. Do you have multiple Steam accounts - accidentally used one to synchronise libraries, but another for step 2?