No one is suggesting birds could be quadrupedal -- or extant birds, in
any case (I made a clear distinction in my prior post).
Japanese shearwaters climb sloping tree trunks using both the wings and
feet, all of which contact the trunk.
I think we can all agree that, if theropods did enter the trees prior to the
advent of flight, then these first arboreal theropods would have been very
ungainly tree-climbers. But they had to start somewhere.
Among extant mammals, tree-kangaroos are clumsy in the trees, and so are
certain species of "flying" squirrel.