People already are being "randomly" killed by cancer, and will continue to be until we develop a cure. There's no threatening left to do.
If I somehow had a choice between N random people dying of cancer every year or N people chosen deliberately by the government dying of cancer every year, and that was the only choice I had (that is, N is fixed), I would choose deliberate choice, with the expectation that we can pick victims who were about to die of other causes anyway and thus minimize the amount of life we destroy. But that reasoning doesn't apply to a fixed number of people being tortured at any given moment.
If instead of dying of cancer you pick something more analogous, like living with multiple sclerosis, then yes my reasoning is similar: I reluctantly choose randomness over government-selected victims.
As for the logistics of battling Dementors, I'm not sure why that's relevant: I've agreed to assume that there's no known way to neutralize them, either for the reasons you lay out or for other reasons.
Update: Discussion has moved on to a new thread.
After 61 chapters of Harry Potter and the Methods of Rationality and 5 discussion threads with over 500 comments each, HPMOR discussion has graduated from the main page and moved into the Less Wrong discussion section (which seems like a more appropriate location). You can post all of your insights, speculation, and, well, discussion about Eliezer Yudkowsky's Harry Potter fanfic here.
Previous threads are available under the harry_potter tag on the main page (or: one, two, three, four, five); this and future threads will be found under the discussion section tag (since there is a separate tag system for the discussion section). See also the author page for (almost) all things HPMOR, and AdeleneDawner's Author's Notes archive for one thing that the author page is missing.
As a reminder, it's useful to indicate at the start of your comment which chapter you are commenting on. Time passes but your comment stays the same.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: