Isn't part of the problem with the so-called paps outside the school the fact that Harry is sending his kids to the wrong type of school? A school like the Dragon, located in a town, would be expected to present security/traffic issues. But it's also a highly academic school for the children of the wealthy professional classes, which is not quite the same as the aristocratic, sporty, country types that Harry is used to. Harry was himself talking about wanting to watch Archie play rugby, which would be more of a priority at a different type of school. I can imagine the Dragon School - if that is indeed where Archie and Lili went for a couple of days - was a bit of a struggle for them, especially when they were brand new to the country.
Similarly, this new school also appears to be too exposed on the street for the type of privacy Harry demands. Somewhere like Lambrook, where William's kids go, with its huge private grounds, "country people" vibe, slightly more traditional outlook and greater emphasis on outdoor activities would have been more suitable. Of course, I don't expect Harry to have sent his kids to Lambrook, but there must be similar schools near where they live.
This whole mess is all because Harry can't be bothered to do the right research, consider the best school for his kids and plan in advance.