The answer is (in Nemo fashion): The Drop-off! We rode Nemo on Sunday, 6/17 and got to Mickey & Friends structure by 6:50 am. Obviously, the structure opens at 7:30. There were about 15 cars ahead of us in line with their engines off, so I decided to do the same. It was at this time when it came to me that "dropping off" would have been the best thing. We could have fed the kids breakfast while waiting at the front gates. Instead we sat in our car for about 40 minutes waiting for the structure to open. So the best case would be, use the drop-off, have someone park the car in the structure and have them either catch up in the line outside DL (if no EE, which Sunday the 24th will be) or catch up in the Nemo line.
BTW, you catch up to your party in the Nemo line up until you make the transition underneath the ride signage. The line does not queue anymore right underneath the monorail station like it used to. It is mostly a staging area that helps the CM organize the groups to load. Just let the CMs standing there know and they will let the driver rejoin your family.
We ended up waiting a little less than an hour for the Nemo subs and we were lined up right by the monorail columns at the gate. If we did the drop-off methodology, we would have easily gone on in the first round of subs.