I don't think this has anything to do with private or public school. Our public elementary school is in the oldest part of town. All residential except for the school, small lots, narrow roads. Probably half the kids are within walking distance, and the district actually offers
free bussing to practically every kid in town, so you'd think there wouldn't be an issue with parents dropping off and picking up kids.
But every year the school has to send out 4-5 reminders to parents about where to drop kids off, where to park to pick them up, to please not block driveways or the road, to stay out of the bus loop and teacher parking lot. They've even tried making the roads near the school one way during drop off and dismissal but unless the town has a cop to spare that day (small town) most people ignore that.
I use the bus and avoid the insanity at the school at drop off and pick up, but on the occasions I need to drive my kids, I come home with a headache. It's nuts.
Luckily they're building a new addition on the school and part of the plan is increased parking and a car pick-up loop, so hopefully the problem will get better. Of course, they also plan to go to subscription bussing so that will mean more people driving instead of using the bus so they don't have to pay (probably me included) so who knows what will happen (and don't even get me started on how bad it is with all the construction going on

!)