Even if true (1st I've heard of it), from where does he get his authority to police, judge, and punish for what happens on private property or public streets?
(speaking descriptively not normatively)
It's called "IN LOCO PARENTIS". School authoritah's have rather broad discretion to "police" (these are not criminal matters, but admin matters, so that term is a bit inapplicable) conduct that happens at the school AND at the school bus stops, which are effectively extensions of the school for admin matters. School bus stops are within their locus of control.
As an example, we had a bunch of kids at local high school do "fight club" after school let out. In one instance, they conducted their fight on one of the school's playing fields. School security responded and the kids were administratively punished by the school. Police were not involved.
They quickly learned to hold their fights on NON school properties (church parking lot, etc.). WA state has no law against fighting in public. It is NOT assault if it is mutual combat (agreed upon by both parties) and thus the private property owners would call police to break up the fight on the property and eject the kids, but that was it. No criminal action was takne, since there was no crime, and the school couldn't touch them since it wasn't school property.
We could break up such fights on public etc. properties under our community caretaking function, but it would be a noncriminal thang . Break it up. take the kids home and let the parents deal with punishment, if any. Our purpose in breaking up the fights was protective, not criminal . Parents appreciated our actions but were pissed off in some cases that no court actions could be/would be undertaken for the fights. If the fights were organized, etc. and/or they charged admission etc. then they could be sanctioned for conducting unlicensed boxing etc. matches RCW 67.08
There have been several free speech cases won by students who posted negative stuff about teachers on websites, facebook, etc. - as long as the postings were done from an off campus location the school could not touch them.
We explained to the school that we could not "punish' the kids for fight clubs, since it wasn't criminal. The school, though can and does punish kids for all sorts of stuff that isn't criminal (as well as stuff that is), but violates school rules of conduct as long as it happens at a school property or during a school function (like a field trip) where school authoritah's retain in loco parentis authoritah.
Also, see the "bong hits for jesus " case which established school authoritah pretty convincingly for off campus school functions (terrible reasoning in this case imo but it is the law of the land)