Birdo is certainly an important character in the Mario series. She was the primary boss in Super Mario Bros. 2, along with appearing in dozens of Super Mario Party, Mario Sports, and Mario Kart games. I would say that she's more deserving than Waluigi for becoming a character, but she's another character that still isn't quite there. Maybe if they remake Super Mario Bros. 2 or if she gets a starring role in an upcoming Mario game, then she would be far more deserving of a slot.
For her move as an assist trophy, I think she could shoot eggs like she did in Super Mario Bros. 2.