How is this at all related to the mafia and anywhere near blackmailling schools?

It would only be blackmail if the service they are gating access to is of significant enough benefit that it causes enough harm to the schools to encourage them to also allow the gemini access, but that's counter to your claim that "nobody needs that company anyway"

Google is a private entity that is allowed to decide how anyone uses their products.

The schools, being entities themselves, are in turn allowed to handle this however they want. Perhaps rolling their own search to encourage learners to rely on better sources. Perhaps blocking AI generated responses. Perhaps blocking search results that advertise products to children.

Or they can just use any of a dozen other search providers.