Thank you very much for the comment and information provided. We will clarify our approach to private roads for such situations. Added tag access = private for roads and gates.

Thank you for your comment. I removed the access tag guided by osm.wiki/Key:access, because the school is a public place, it can be accessed not only by personal permission, which is confirmed by the driver’s report. An access=destination tag has been added to the gate to indicate that access is available here for those associated with this place.
Access=private means that you can’t get at this location without direct owner permission. If postal service, delivery trucks, guests or clients can reach these roads as their final destination, access should be other than private.
Please share if you have any additional information on restricting access to this place.

Thank you for your comment. Yes there is, the footway has already been added.

Thank's for your comment, I added pedestrian connection to this road based on customer feedback and mapillary images.
[changeset](osm.org/changeset/74791878)