I've not totally grokked the Birdy front suspension geometry, but I think it may incorporate some anti-dive under front braking. Under other conditions, with no torque acting in front, the suspension will move as do all suspensions, that's the intent. With a relatively short wheelbase, it's gonna bob a bit if the suspension is sprung soft, but that's a big qualifier and can make a huge difference. Certainly a bike with front and rear suspension would be better on rough surfaces unless you have huge cushy tires.

Sounds like the OP is reevaluating their current bike and perhaps a refresh of it.