Kuvult,
I would also add that you don’t have to belong to any of the movements at all. Yes, there is a minhag of davening and wearing a kittel, but there is no reason why one needs to pledge allegiance to one way of learning and thinking. This is devaluing Torah and making it into a tool of building group-think.
Maybe we are assimilated into American freedom of association and “factions” leading to a free market of ideas, so we also divided into groups competing ideologically. In these terms, separating by the style of the hat leads to market segmentation, every movement controlling their own institutions, same way Apple controls macs and iphones. From the POV of consumers, if parents see more schools acceptable hashkafically, there will be more competition and more value for parents, as intended in halakha (competition is limited in regular trade, but unlimited among teachers and perfume peddlers),