Then why are they agitated about being banned from them?

I think a club can have pretty much any kind of rule, no matter how silly. They can ban all bikes except red, for example. Then it's up to the riders to decide if they want to be part of a club like that.

I think banning tri bikes per se is pretty dumb. But then some roady types (including triathletes) seem to enjoy playing some rather childish petty games with one another.