Due to recent changes in Electronic Arts Login system, you need to perform the following steps in order for the Origin License management to login your store accounts without showing the login screen.
Failure to perform the following will cause the Origin License Management to fall back to keyboard input, which is not 100% foolproof and might fail under certain circumstances.
NOTE: The following steps need to be performed FOR EACH ORIGIN ACCOUNT
1. Open https://www.ea.com/
2. If you are already logged in, sign out using the icon on the top right of the webpage
3. Sign back in with your Origin account
4. You should be greeted with EA's Term Agreement. Accept the agreement and click "NEXT"
5. If you are asked to verify your identity, click the "SEND CODE" button and complete the verification
5. Done. The account should now automatically login without showing the Origin login screen on the client.
Repeat the steps for each Origin store account you have.
NOTE: It is possible that you will need to repeat the above steps again if EA changes its license agreement terms.