I think the reason why people didn't like Shelby was the way she went about her ambition. Ambitious women is not a problem. But going behind the back of people in the chain of command is.
Riker made it clear he would be presenting all options, which included hers, to Picard right after the briefing. Shelby instead did an end run around him with the express purpose of trying to look like the best candidate for an XO.
That's not respecting the chain of command, and that is ultimately why Riker was irritated at her in the turbolift. And rightfully so.
If anyone under your command, man or woman, did that, of course they would be looked at badly. It also shows that the person's ambition outweighs following orders, and that is definitely a problem in such a setting.
(And to be clear, I never saw Shelby as a villain. Far from it. She brought some spice to the career of Riker and the situation. That was much needed. But she let her ambition override her training and better judgment.)