Gotcha. You just annoy the people who live in the next road over.
The best way to annoy the people in the next road over would be to maintain the status quo.
This is due to the long known (and carefully ignored) phenomenon of induced demand: the easier we make it to use a certain form of transport, the more people use it; or as one person put it "You sow roads, you reap traffic". In the case of schools, if you make no attempts to reduce cars at schools, then more people will drive and the problem will soon spill into the next road over because the road by the school is full.
The reverse has also been observed due to Induced Demand; making it harder to drive somewhere will reduce the number of cars making the journey. In thise case, making it harder to drive close to a school will mean less people do it, especially if you make it easier to get there in other ways.
For example in our village school, making a one way system and banning non residents cars on certian streets, while making it possible to cycle and walk direcly, has the potential to reduce traffic on school runs, just because people will usually take the easier option.
In practice, you can either benefit car users or benefit everyone else; it's a zero sum game. Fortunately this means that by making it less convenient for drivers you automatically make it easier to use other forms of transport.
We need to think more creatively to get rid of cars, not dismiss new ideas without trying them.