The alliance meant different things to different people in the Klingon Empire. The Klingon leaders had respect for the Federation's strength and honor and knew they needed them as an ally against the Rommies but could gain political points by badmouthing them.
And Gowron would have never broken the alliance in DS9 if shifter-Martok hadn't been pissing in his ear, feeding him misinformation, and getting him to do dumb things he knew the Federation would never go along with.
Klingons who aren't being manipulated by somebody else may want to distance themselves from the Federation, but they certainly know they benefit from it.
He sure would have, if she had spent the last several minutes chasing Nikolai through the corridors.