When I was growing up kids within a mile of the school would walk themselves to school and back home with very few exceptions, often starting with first graders. But even a half mile is a long enough distance that many busy parents in the suburbs would choose to drive it rather than walk it four times a day.
Then sometimes, they leave their car and go in the school and we are trying to home or leave and they get angry with us!
Parents: It's never ok to block somebody's driveway! Residents: It's not illegal for people to park on the street in front of your house (assuming they aren't blocking your access of course)
And if you want to just do a drive by pickup, sure, no worries, stop in the middle of the street (with cars either side), and have your kids younger sister hop out, leaving the car door open, and go wander down the slope to help her brother with his equipment and carry it back up the slope. No-one else is using the street or lives there, after all, so they can wait a few minutes. Bonus points if you get belligerent with them or flip them off for being "impatient".
The parents that arrive early but have kids that are not ready to be picked up is precisely why the queues get out of control. At least with this method the cars are moved along much faster. If a car arrives in the queue but the kiddo is not yet found, when the car reaches the front it is asked to move to a holding spot out of the queue--much like fast food places do when your order is going to hold up everyone else in line.
And the roads are limited by the school's ability to acquire land.
And no reasonable school is ever going to set aside any more land for a car line than is absolutely necessary that could otherwise be used to provide more school building/field.
The system you describe isn't necessary to cause the traffic. It's not the root cause.