Yes, the mistake was entirely mine.
> As for why Birmingham in particular, I don't see it as some kind of gotcha to say because ...
Fine, that's a perfectly valid reason in itself, but the West Midlands police did keep quiet about that being the basis for the ban, only saying so (in far less detail than you) after the match had taken place: https://www.bbc.co.uk/news/articles/cqx3d5enx0xo, which in itself seems suspicious.
If your point is "hooliganism happens and is treated on a case-by-case basis, in some circumstances warranting bans" then my response would be to agree, though I still hold that such hooliganism is rare. This isn't the 1980s any more. Any further dispute is about the facts of the particular case, and what I've seen from the Commons committee which questioned the West Midlands police chief doesn't fill me with confidence that your interpretation is the correct one.