"No, they mean any Constitutional rights"
So knives and guns aren't banned? Drug searches without consent are ok?
"To the extent that that is true, that’s not a waiver by the student. But while it allows considerable regulation, the limitation on free speech on schools is quite narrow."
It seems quite broad based on the case laws they cite. It can restrict vulgarity, imaginary stories containing violence, prayer, and basically any topic they feel would cause a disturbance, which is a lot.
"Lenity is not absence of strict definition of what is prohibited, but government’s power to not fully prosecutr what is prohibited. Strict definition is about the boundary of what may be enforced."
If a law is defined using vague terms, then any ambiguity is supposed to benefit the defendant. The reasoning is that you can't expect people to abide by a law that is unknowable because it lacks a clear definition.
"then the solution is for the government not to run cheerleading teams, not for thr government to assume unconstitutional powers."
"That is not the basis for the (fairly narrow, in the case of free speech) power of government to regulate conduct in schools beyond what it can do in other contexts."
Exactly my point about education in the previous comment. These exceptions to rights that are necessary for the school to function, by your own argument, make it so that the government should not be running schools at all.
You reference these narrow restrictions from previous rulings. On what basis where they created? Not on some Constitutional text, but on the court deciding those rights can be restricted. So too could they rule in this case and cement a restriction that was not challenged before.
"That is not the basis for the (fairly narrow, in the case of free speech) power of government to regulate conduct in schools beyond what it can do in other contexts."
Then what is? I see nothing in the Constitution allowing for those restrictions on this right. They were manufactured by the court through common law customs/beliefs and cemented in case law.