Of course they count.
My assertion, my reading of what they were stating in that tweet, is that the function of a school isn't healthcare -- you go to a doctor for that. You have nurses for minor stuff and emergency services like using the defib, but that's just ancillary help. A school offers incredibly limited services.
You're not relying on the school to for your child's healthcare. Just like you wouldn't with mental healthcare.
I'm guessing they were complaining about schools intervening in a way they wish they wouldn't where mental health care was considered.
I mean, imagine your kid comes home from school with insulin, a weight loss plan, and a diabetes diagnosis. And they tell you they got their fist shot that day. That would be a violation of your expectations of service a school ought to be providing. You weren't in on it, they decided a bunch of stuff and didn't tell you, and they acted on it.
What you might expect is that a school nurse would tell you about issues your kid has been having; or take care of a scraped knee or this or that. Or that the school counselor might fill you in on what he's noticed with your kid. You don't expect these people to be making important decisions for you.
Make sense?