Some of these functions are not intuitive: maybe they regulate the function of another protein, maybe they only function in the context of a particular stressor, etc. You can think of nearly unlimited scenarios to apply and you start to understand the complexity of understanding how a gene functions.
Function is a rabbit hole. Biologists get in big arguments about the semantics of this all the time (http://cryptogenomicon.org/encode-says-what.html). I don't really care. I care about the minimal set of necessary proteins for a model organism to exist and reproduce in a media-rich environment. And, whether there are actually subsets of mutually compensatory groups of proteins instead of a single minimal set.
Protein function is one of those things that, at first, seems really simple to define, but the further you go down the rabbit hole, the more complicated it gets, until it's fractally complex and you realize that not only does the exception prove the rule, it's all exceptions.
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?”
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.
For the benefit of readers unfamiliar with the field: this is wrong.
You can show this experimentally. Construct a gene that produces a non-human protein and introduce this to a human cell/genome. That gene would not have a “function” but still exists in the genome. This is actually happening all the time. Some viruses integrate their genomes when infecting cells. Viral integration is one of the factors that shapes genome evolution.
Your viral integration example is actually a perfect example of one where all genes indeed do a have a function, but they are not readily apparent to us. Genes that control latency may not expressed until specific conditions, and that is their function, to control expression. Some genes control integration.
I spent 5 years of my life doing my PhD studying viral replication, and the "unknown function of viral genes" was a constant topic of discussion, but we all agreed, they have a function.
"Taking up space in the genome is a function" is a great example of this. While I'm sure you can find examples of "spacers that when deleted are fatal", the fitness effect of protein-coding regions that contain no utility is still an area of research. To me, functionality requires selection, although that's probably not necessary or sufficient!