The grinding is likely to be from what is called the sled motor. If I recall correctly, these use a small motor that drives a screw shaft along which the lens assembly rides. You may have a defective gear on that motor. These gears are made from nylon and teeth can break. Some units use belts between the motor and the screw shaft but these I believe use gears. Since these move only a little at a time for CDs, the defect may not be severe enough to affect normal play. Games however tend to require data from random places on the disk which will require extended (and quick) use of the motor.
If you can get access to the drive itself, post some photos of the mechanism and we can confirm this.
Dan