The people handing down the aforementioned decisions were elected specifically to do things like this. That doesn't outright make it authoritarianism. (Yes, most elected US politicians on both sides of the aisle skew fairly authoritarian, but that doesn't mean the process itself is authoritarian)
I prefer to think of these words as points in a spectrum, rather than definitional prisons with tight boundaries.
Allowing school districts to do what they are constituted to do, is more democratic than delegating it to the State governments, which is more authoritarian.
Authoritarian is also not an antonym of democratic. They correlate, but not as strongly as any advocate for democracy would prefer.
Actually not so; it is only centralised. The democratic process is pretty clear that the people want education decisions handed down by an authority "at the top".
A decentralised approach to eduction would let parents make on-the-day decisions about where to send their children and probably have some sort of market-based approach to how education was funded (maybe even parent-pays). That approach has been mostly rejected politically.
The correct decision was reached, in time (before the start of the school year) for it to make a difference.
Tons of people and thousands of schools vigorously debating is the democratic process. A decision handed down from "the top" is authoritarianism, and as we've seen repeatedly this year, authoritarian decisions are resisted even when they're the right decision.
For something like masking, you can justify it with the urgency of the situation (even though I'm compelled to add that "the top" got this one wrong during the crucial months of February and March). For starting school in the fall, you can't.