I don't have this problem but I have a suggestion on what it might be.
Maybe the iPod port on your iPod is worn and losing connection while you're listening to music through the player? If the system loses connectivity, it'll have to wait until it re-gains connection to play again. Since the player will play music and then shut off at a random amount of time after start-up, this makes me suspect a connection issue.
You might not be able to see the connection on your port degrading over time but contacts will wear after a while.
If you have a friend who has a player, try plugging it in and seeing if the music will play without problems. If the same problem exists, then it's in the car side, not the player side of the system. The car connector could be bad at a number of places in this case.
Given that not a lot of people are having the same issue, it could be only your system with this specific issue. The next time you plug the player in, see what happens if you wiggle it a little - not too much because you can do damage to your device or the cable connector but give it a gentle tap near the connector and see if the device goes 'mute' on your or comes back if it's already muted.
Good luck figuring out what's going on! These issues can take a while to figure out but if you grab another player and see if the same problem exists, that can help eliminate where the problem might be.