I think the semantics are largely dependent on the level of abstraction you are reasoning at. Here I use abstraction loosely to refer to conceptual or physical scale / granularity / resolution. For example, we are (probably) both using the term “gene” as shorthand for the concept of “gene or gene product(s)”. Likewise the term function can refer to phenomena occurring on molecular, cellular, etc length scales, or more amorphous groupings of phenomena such as “flux” through a biological pathway.
But if you really drill down into the nitty gritty, then the “function” of a “gene” is its complete set of state altering / modifying relationships with other bio entities. In this sense, all bio entities have functions because they all have functional relationships with other bio entities.
So yes, all genes, pseudo genes, isoforms, etc have a “function” even if it is redundant, taking up space, or just soaking up some of the pool of tRNA.
Also the minimal genome stuff is pretty fascinating! One of the best research questions I’ve ever heard was, “what are the essential genes of unknown function doing?”
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.
But if you really drill down into the nitty gritty, then the “function” of a “gene” is its complete set of state altering / modifying relationships with other bio entities. In this sense, all bio entities have functions because they all have functional relationships with other bio entities.
So yes, all genes, pseudo genes, isoforms, etc have a “function” even if it is redundant, taking up space, or just soaking up some of the pool of tRNA.
Also the minimal genome stuff is pretty fascinating! One of the best research questions I’ve ever heard was, “what are the essential genes of unknown function doing?”