I've done rope drop multiple times at each of the parks, we've always driven, and I've never had any issues with traffic flow. At MK you might have a 3-4 car wait to pay for parking at the TTC but that's about it (and btw if you are going to the Contemporary you don't have to pay the $25 or whatever it is at the TTC booth, just tell them you're going to Contemporary for Chef Mickey's). The only time I have parked at the Contemporary and NOT had reservations for Chef Mickey's, I tried to do valet and flat out told them that we wanted to park and walk. They looked at me like I was stupid. They were like uhh, just go park over there then, why are you paying us $25 for valet? I said uh, I felt like I should pay for parking since I didn't at the TTC booth. They again looked at me like I was stupid.
I should add it's been a few years (before Rona but < 5) so things may have changed.