In general, this isn't true at all, since nowadays there are one time use CD keys that lock onto your account. If EA still runs this way, then I might as well buy the digital version, but that sucks, and EA is still in the past.
Are you talking about PC, or console?
If this is PC, then, yeah, you don't need the discs to play, since you're installing off the discs and launching out of Origin.
I assumed that you had meant from a console.
Mass Effect 1 on PC used the controversial SecuROM software which meant no disk was required.
I am pretty sure that after 5 years and having removed the controversial software and implemented a new scheme, they are still able to let you play without the disk installed.
I also recall reading a way for you to add your "Physical" copy of Mass Effect 1 to Origin.
https://help.ea.com/article/redeem-your-serial-code-in-origin
