Yes, if you have a car park on private land, no matter how long it has been there, you will have to pay the levy on the spaces.
In the specific case of a school, if you require your staff to be able to transport many kg of books, busses become much less appealing. Many bus routes pass the school, but you may not want to struggle with the books onto busses crowded with kids you teach every day, and they are frequently delayed (eg my daughter regularly waits half an hour for a bus after school). There is a significant issue with routes in that they tend to go to/from the city centre, but very few go from one area to another, but that's a whole 'nother discussion. Some of the local schools have great cycle infrastructure, some (like my wife/daughter's) is much more hit and miss (and the last mile from our direction has almost none).
In principle, I think it's a good idea, and I've not seen the other details that go along with it. I fear it will make recruitment for the schools and hospitals harder, and it will have almost no impact on traffic as much of that is driven by students, parents, and patients, and this won't affect them at all.