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!