Birds did eventually find a way to get the forelimb involved in
locomotion again, but no quadrupedal stance.
One minor point, which doesn't really detract from your statement, which is
essentially correct given that no modern bird uses quadrupedal locomotion
when on land. But the first birds (like _Archaeopteryx_,
confuciusornithids, and so forth) probably used all four limbs when climbing
trees. Thus, they were arboreal quadrupeds and terrestrial bipeds. The
same is probably true for microptorans and _Epidendrosaurus_.
Among modern birds, juvenile hoatzins are arboreal quadrupeds. Certain
shearwater species have observed using their wings when climbing up tree
trunks.