Searching for information about fish online has been enlightening. There's an awful lot of information out there, but it's often hard to gauge reliability. I realise that shouldn't be a revelation to me, but the truth is that I have rarely looked "blind" - I usually either have a sense of what I am looking for, or I have restricted my search to sources I feel comfortable trusting. This hasn't been a viable strategy this time.
I initially got the impression that ich was a water quality issue - that the causal organisms were normally present in aquaria, but only became a problem when fish were otherwise stressed. That proved to be a misconception - Ichthyophthirius multifiliis appears to lack a dormant life stage, so it can only be transmitted by infected fish or by plants (or other substrate) upon which the tomont has encysted. The infectious trophont is the sensitive stage - this is the one that can be affected by medications, and it appears that the infectious trophont stage can't survive more than a couple days outside of the body. Hence the suggestion that they best way to "clean" a tank is to leave it fish-free for a few days. Of course this doesn't answer the "Typhoid Mary" question - are there outwardly healthy carriers of the disease? To some extent this is true - since the parasites tend to infect the gills, it's possible to have an infected fish that lacks the characteristic white spots. But these fish will still only remain infected until the trophonts mature and drop off. Are there dormant infections - ones that will remain in the trophont stage until the fish is stressed?