I thought my mini's optical drive had failed as well. It wouldn't mount CDs, DVDs, or anything. It'd just sit there and spin up and down before spitting it out.
A dirty laser lens was the culprit. I took a cleaning cloth I use for my glasses (really soft, yet very thin), poured some 91% alcohol on it, wrapped it around a credit card and gently worked it back and forth through the DVD slot. The lens is pretty close to the front, and after wiping the cloth over it a few times, I fired the machine up and tried some media. Everything mounted flawlessly.
This could help your stuttering playback as well. It sure can't hurt to try.