Thomas, you are absolutely correct that the the CD drive is married to the motherboard. The fact that it reads DVD's and not games is generally an indication that either the board has a problem or the drive was spoofed.
When I first got my Xbox 360 I decided to hack my drive so that I could play burned games, and to do this I purchased a second drive for the system and spoofed the firmware from the original drive on the new one. It worked well for about 3-4 years, but one day my system did an update and all of a sudden I had difficulty getting these same games to run. I ended up having to eject the tray many times before i could get it to read burned or genuine games, and then install the game to the drive so that I did not get read errors when I did manage to get it to open, after a while I got sick of having to eject the games constantly, so I simply put the original drive back in and started playing genuine Xbox games, and have not had any problem reading those. The whole time I was having the problems, it never failed to play a regular or burned DVD, only games were the problem.
End Story Time :(
I suppose what I am try to get at here is that your problem to me sounds like the drive was spoofed. If you did spoof the drive, and you did keep a copy of the original firmware, you can undo the spoof and make the drive just like it was when you purchased it.
If you did not spoof the drive, that would mean the CD drive board is failing, in that the chip that would authenticate with the motherboard is going bad. If this is the case, I would consider just going to either a used Xbox 360 or to an Xbox One.