The antimodernism thing is like the least charitable take one can have on Nietzsche but at least it’s not one that’s based on his sister’s stuff.
Some quick thoughts:
His stance on democracy has to be understood in the context of its days, much less developed than now, and in the Kaiserreich also very much class-based, ruled more by mass psychology than consideration of what actually good politics would be – on both sides, though I won’t deny that the nobles and bourgeois of course needed their wings clipped. At the same time he’s very much an elitist in the sense of, erm, personal improvement, sees the need for the individual to transcend the forces acting around them and develop their own path as sublation of everything, contrast that with the political forces in parliament being not even close to that but simple thesis-antithesis with no sign of actually starting to go beyond that and you have an easy case for “Nietzsche simply didn’t believe in the process democracy”.
To all this he prefers “hierarchy” [Rangordnung]
Is that really the term Anglos use as a translation. “ranking” or even “precedence” might’ve been a better choice. Honestly just translate it literally: “Rank order”. In any case and I won’t dwell on it: Nietzsche always describes these rank relations as in flux, not set in stone, and makes fun of tying it to inheritance. I don’t see him at odds with Bakunin, here, who will readily bow to the authority of the bootmaker.
At the same time he warned of the dangers of not having such a thing, of insisting on some moral-metaphysical notion of inviolable human equality, and we just recently had the opportunity to see that kind of thing in action: I’m speaking of the masses of people unwilling to bow to the authority of virologists and epidemiologists, going “nu-uh I did my own research”, meaning they read some bullshit blog somewhere. Nietzsche himself might’ve rather thought about the Jacobine terror and stuff.
Overall, when reading Nietzsche I recommend starting with Thus spake Zahatrustra, as a work of philosophical mysticism, get to grips with what it means for the individual mind, and interpret the rest in that light, and specifically consider whether he might not have framed a lot of things very differently had he witnessed Nazi Germany.
A parallel which comes to mind here is Plato, who likely would be similarly at odds with the modern scientific method as Nietzsche is with the democratic process, stressing the importance of intuition as to not de-humanise the process: Are we, as peoples, really engaging in democracy, or do we let a system of mass psychology rule us?
Lastly, my psychological armchair: Was he someone who often felt alone in a crowd? Yeah, probably. Clowns to the left, jokers to the right.