OK! Here are some pictures of the fix. I wasn't able to find any pictures of the inside of the drive at all when I was doing my initial Google searching so this should be very useful for someone who might have this same issue. To get to the rollers, simply unscrew the four Phillips head screws that hold the gear train assembly in place; the entire assembly should come off.
The piece is basically just a linkage for the two rollers in front so that the rolling action is distributed (only one of the rollers is connected to the rest of the gear train). Without the piece the roller shown on the left wouldn't move, causing more strain on the right one to push the disc out.
It's pretty darn loose fitting and I can see how it might pop out easily if someone grabs the disc while it's coming out or if someone pushes a disc while the PS4 is feeding it in. I just had to pop the left roller out, insert the plastic piece in between, and pop it back in.