The way I see it, the rightwing strategy with the "guns for teachers" push is to make inroads into one of the last, and maybe politically most influential (and usually backers of Democratic Party candidates) unions: the teachers union.
By pushing for teachers to carry guns, the right is also pushing for the hiring and retaining of teachers who might not merely be willing to carry a gun into their classrooms and schools but might also be against gun regulations and ultimately susceptible to taking a position against their own union's policies and political agendas.
Which goes hand-in-hand with the right's decades old strategy (back in the '60s I wrote about its inception under Nixon and Agnew) of delegitimizing education in general, especially the public school system, and higher education in The Liberal Arts and Sciences.
The power of a strong education in a public school system that is accountable to public servants, and extending that to any kind of higher education other than the conservative based kind, frightens those who finance rightwing think tanks and propaganda mills.
A teaching force amenable to tax money going to fundamentalist religious schooling, and charter schools controlled by corporate interests, and schools allowing teachers to carry guns into classrooms, is their goal.