Hey, this is my niche - I wrote what was once the leading policy management system for K-12 districts in the USA. It no longer holds that position, but it originated the niche, and it really taught me how policy is created.

Talking to the school board is absolutely a key piece of that, so that was a good move, but talking to the superintendent is an even better move. Most districts delegate the policy work to the superintendent. Most superintendents consult with legal firms who specialize in educational policy. At the same time, most school boards cannot sit down and have a discussion with you about a topic - everything they do has to be in the public eye, so these little blurbs at meetings are the limit of feedback, unless you can get them to put an agenda item at a meeting to have a longer discussion (Which you may be able to do when specific policies are up for reading or votes).

The workflow of how policy really works is that the school board (while being the actual decision-maker) is the last step, not the starting point. The actual starting point is in your state legislature as they set the state-level requirements. Attorneys will keep up on that and let the superintendents know what is needed to be in compliance with changing laws. Superintendents do make their own decisions, but will consult with policy consultants and recommend policy changes to the board. The board will have the policies presented at a meeting, provide time to read them and give feedback, and then finally have a vote on them.

In terms of influence, if you really want to invoke change at the policy level, start at the bottom of that workflow. Talk to your state legislature. Talk to your superintendent. Respond at board meetings to specific policies. And sure, give general feedback at a board meeting. They all help.

(Of course, not all districts follow that model... but most do.)