I can see both points of view..
I can understand the frustration of the school and the residents about the congestion and dangerous parking, and there is really no excuse for illegal parking at all.
But then, I have been there at times in the past, trying to drop off and rush off to work, and it is really stressful. (Having said that I always found it easier to park 5-10 mins walk away rather than try a get nearer the school, and it did not add much time on, and was less stressful).
I do think closing the road will just move the problem to the next road that is open.
I do think some schools could do more to help, with earlier drop offs or organising drop offs at set locations and walking together.
Some schools near where I work, in a more deprived area charge £1 for breakfast club and they can drop off from 7:30. Our school does not have this. There is a private nursery next door which charges £9 to drop off at 7:45 and £7 to drop off at 8am daily (no breakfast) which is not really affordable, and we just use in emergencies.
Pre covid, we relied on grandparents 1 day per week, and otherwise worked our hours around the drop offs and pick ups, sometimes we helped friends and vice versa.
I think if all schools could provide the same that would be a start...