Not usually. Since they now allow guests into the park up to the hub prior to park opening there's not much if any advantage to it. If you rushed through breakfast you might be near the front of the entrances of each land. Be Our Guest has an advantage since it is beyond the holding area.
Only if by "if you rush" you mean eat in about 15-20 min will it get you near the front of any entrance. There will be people lining up at those ropes by 8:15 and by 8:45 (which would be my definition of rush), there will be quite a crowd at each of them.
OP if you want to eat at CP, make an ADR for around 10:30 and enjoy the early morning touring attractions.
You cannot enter park early like a ppo bog breakfast can get you. It’s as helpful as eating breakfast before coming to the park in the sense you don’t need to stop to eat at 1030 or something.