The local township library and school board are both literally the government, and the book banning is largely happening based on local election results (typically county positions and/or school board positions -- both from election ballots)
It's not wrong at all to say the government is banning books -- that statement is objectively correct, it's just happening at the lowest levels of government.
> I assume you mean the state or federal government wants to introduce a new law.
Why? If it were any other issue, no one would care about the distinction. If Allegan County, Michigan banned firearms, people would still claim it's an infringement of 2nd Amendment rights, even though it's just "concerned parents asking relevant institutions at the local level" and not necessarily a state/federal law. But when that same exact county bans books and/or libraries instead, suddenly it "doesn't count" for some reason?