Anonimity is of no importance at all, it is simply an unfortunate result of the lack of continuity in collecting. We have snapshots of individual singers, rather than a historical record over time.

If a collector of Harry Cox's songs could also have collected the versions from Harry's sources, and from their sources, all the way back to the original composer (whether broadside hack, stage show, ploughman or pitman) then we could observe how these songs evolved over time and how each individual contributed to that evolution. Instead, we can usually only observe how they evolved over space: we can see that different versions of the same song arrived in different locations, but not how they got there.

If modern research is able to uncover the identity of the original composer of a song which has subsequently gone through the folk process, that doesn't suddenly invalidate its credentials as a traditional song. The only reason why recently composed songs usually fail to qualify is that they haven't yet been around long enough for the variations to arise, especially as most singers now have both the mental concept of a "correct" version and the resources to find and learn it. However modern songs can be subject to the folk process and do enter the tradition, and as another thread demonstrates, this is even easier for tunes.