Treat charter school teachers like public school teachers probably starts with acknowledging they are in fact public school teachers. That would be preferable to constantly marginalizing them, and their schools, as nonpublic entities.
State laws are what forms charter schools. They are accountable to the state based on those laws. Last time I checked, those laws are written by people who are democratically elected in their states.
They NLRB case that you mention, and the many union-influenced commentators pointing to it as definitive proof charter schools in general are nonpublic, is partisan (Democratic) and part of a paleoliberal position that seeks to preserve old world ideas about how we organize public education. That isn’t progressive or particularly promising for the future.