Let's keep the loaded semantics out of the picture, shall we?
Homologies are by definition shared traits from common descent. So you cannot theorize about homologies not being shared traits.
You can discuss similarities that are considered to stem from common ancestry.
Not trying to be picky, but think the whole discussion can proceed better if we keep such terms out of the discussion.
Otherwise, we really have to decide if similarities should be called homologies, homoplasies, or neither (which is of course what we are trying to do here, but just to make a point).
Moreover, for convergent evolution to take place similar selective pressures must be in evidence.
That seems logical except that convergent DNA sequencing suggests otherwise. But it's an interesting point. If environmental pressures must be applied to create convergency, and we find examples of convergency that do not seem driven by environmental pressure, would that be evidence of a hidden environmental pressure such as an intelligent agent?
Why do birds show so many homologies with therapods when their lifestyles are so different?
Good point except we need specifics, and I have a question, are not the homologies advantageous for both groups of species? Assuming they are, that they are due to natural selection, why could they not be an example of convergent evolution?
Can a trait not be selected for, even when the 2 species have different lifestyles? You are assuming that the trait can only evolve independently from one set of circumstances, and that's just an assumption.
Plus, what if it did not evolve from convergency from outside selective pressures, but from convergent DNA, or ID (an intelligent agent), or just by golly, the same trait worked for both and independently evolved.
Common descent is not the only answer here.