Years ago I purchased ME on CD. I did not buy any digital download version. However, the digital download version appears in my Origin library. Similarly, ME3, which I purchased on DVD, is visible in my Origin library.
The difference is this: The ME icon does not launch the installed CD version. The ME3 icon does launch the installed version along with all purchased DLC. Why does ME appear in my library if it cannot launch the installed CD version like ME3 does?
Also, as BDtS is now free can you add that to the Origin library along with Pinnacle Station? Can you add BDtS and Pinnacle Station to the Added Content tab for ME in the library?
I guess you have registered both games with your EA Account and they are now available in Origin for your. Have you tried to install Mass Effect 1 from Origin client right away, without using your disc? And in case you are missing any of your items, you should bring this directly to EA Advisors. https://help.ea.com/en/contact-us/
I am not entirely sure if you already have the DLCs installed, if not you can get "BdtS" and "Pinnacle Station" here.
It is not necessary to add them to Origin because they will start automatically if you start the game.
If you decide to download the game with Origin the lunch over Origin will work and the DLCs will appear in the "Extra content" section of the game.
You could also try to repair the game. (in Origin > Game Library > right click ME1 > Repair)