Hey folks, I had the same issue for weeks, tried all the same things (restarting, reinstalling, using ea app recovery, clearing cache, uninstalling an manually deleting all reg entries and folders, etc) and of course none of them worked.
FINALLY managed to fix the issue and am posting here as a sort of public service announcement. The EC10005 error seems to have 2 main causes:
1. EA Server Issues. If this happens nothing to do but wait, but in reality this happens very very rarely and only for a short time.
2. Interoperability between the EA Desktop app and Origin. THIS IS THE MAIN CULPRIT and is due to some pretty stupid programming.
Essentially a fresh install of the EA Desktop app NEEDS Origin to build out a games index. Once this is done, the EA Play app takes over. The app doesn't explain this in any way. It will simply throw up the EC 10005 error if it's unable to grab the database from Origin.
So to log in to the EA Desktop app for the first time, you need to take the following steps (some steps may be unnecessary, but this worked for me):
0. (Optional) Uninstall Origin and EA Desktop. Restart PC. I did this to troubleshoot from a clean slate, but this may not be strictly necessary.
1. Download and install Origin.
2. Login to Origin with your EA details.
3. Open Origin and confirm you can access your library of EA games.
4. KEEP ORIGIN OPEN (this might be unnecessary, but it worked for me so just do it for safety)
5. Download and install the EA Desktop app.
6. WITH ORIGIN OPEN AND LOGGED IN, open the EA Desktop app for the first time.
7. You will get a popup message asking you to close BOTH programs and then reopen EA Desktop. Do this. (make sure they are completely closed, not minimized to task bar)
8. Once both apps are completely closed, wait 10 seconds, then open the EA Desktop app.
9. You will be asked to log in to the EA Desktop app. Enter your login details and THIS TIME your login WILL WORK and the app will load correctly. No EC10005 error code.
Clearly it's ridiculous the app has been designed this way, but at least we now have a workaround. Onward and upward fellow gamers.