[dekhn]

The problem is that proteins do some things passively unrelated to (say) their enzymatic ability. Is that secondary functionality a function? What if it's binding a molecule, but then releasing it before the catalysis occurs (wasted effort).

I mean, I know a person in grad school who worked on finding the function of a protein for a long time. It was given by a collaborator and had high sequence similarity to a known enzyme in a related species. She tried every possible functionality test to see if it was a protease, or any of a hundred other enzymatic reactions. Eventually it turned out the collaborator had mistakenly given them an alanine-scanned protein with the necessary functional residues replaced, so she never detected any activity because there wasn't any. Does that mean the protein had no "function"? It was binding water molecules, even plausible substrate, but just never helping a transition state form. Even if you replaced the working version of the protein with the broken version in an organism, if it wasn't a completely necessary protein, it would continue to reside in the genome with no function for some time until (perhaps) neutral mutation due to lack of functional selection caused the protein to be non-expressed and it starts to rot away into a pseudogene.

The main problem with your research question is that it still hasn't been completely resolved- there are proteins remaining which are necessary, but their functions are unknown.