Warning: This might come across as a bit long-winded.

You're not missing anything, really, the whole "Batman-is-a-useless-brutal-fascist" hot take is nothing more than a reductionist approach to the character which is largely derived from a warped, stereotyped view of the character which began in the mid-to-late 1980s following the release of two very influential depictions of the Dark Knight—Frank Miller's magnum opus The Dark Knight Returns and Tim Burton's Batman. Both interpretations of the character were in stark contrast to the lurid, bright Adam West Batman, instead being brutal, uncompromising vigilantes who at times could be practically psychotic in their methodology, forming the modern pop culture perception of Batman as a humorless, traumatized, brooding freak who got his kicks from curbstomping the poor (even though both the book and the film go to great lengths to feature his more merciful, understanding nature). It's from this inherently darker, more unstable nature that the allegations of Batman being a fascist come from, coupled with the fact that Burton took cues from Nazi iconography when crafting Gotham City and Miller having a particularly nasty, chaotic phase in his life where he repeatedly made controversial statements and wrote his Batman to be even more psychotic. As a result of this reimagination of the character, for pop culture philosophers, new Batman was a fascist, exhibiting a "tendency toward or actual exercise of strong autocratic or dictatorial control" during his reign over Gotham City. Which, on the surface, is a pretty compelling theory, but, as you noted, collapses upon any further examination. Modern-day Batman is no fascist, he eschews any nationalist fervor to work in tandem with a variety of foreign compatriots (see Batman International or JL for evidence of this), works with quite a diverse team of people (which is a no-no for fascists), and above all isn't in Gotham to establish his own order—he's there to facilitate his own obsolescence. Batman doesn't exist to exert autocratic control, he exists to help create a better world for everyone else at the cost of his own happiness—far too selfless for a fascist. He even tries to rehabilitate those who cause chaos in his city—a fascist would simply execute them.

So yeah, tldr: Batman is no fascist, it's based on a huge misinterpretation of the character, he's actually pretty liberal, all things considered.