For example loss-of-function mutations can be detected without resorting to other genomes beyond the reference. Inversions, translocations and copy number variations can also give clues to illness.
Source: I am the lead web developer for uniprot.org, and I deal with lack of real info daily as does everyone in the Swiss-Prot team.
Am I right to say that the bandwidth and storage needed to upload whole genomes is prohibitive?
Bandwidth and storage are infrastructure issues that could be solved with enough money. In biology we are still lacking good technical solutions to do the chemistry and even something "simple" like getting a protein crystalized so we can determine its 3D structure is not trivial or cheap yet.
Fast genome and RNA sequencing are massive improvements and really help. But the basic understanding of what all those genes, regulatory and other parts do is still relatively primitive.
We often "complain" that we wish that people stopped sequencing and went back to do more biochemistry for functional characterization instead ;)
Of course, not every mysterious illness can be solved this way...