Thought-provoking post, Adam.
It seems to me, though, that introducing explicit randomness is only necessary if people (in social decision process) prefer a lottery to some other version of randomness. In most countries, access to health care has significant randomness associated with where you happen to live, who you happen to know, and how you talk, act and look. The more interesting question to me in your post is whether the explicit lottery is better than a CEA approach. And here, you haven't convinced me precisely because of the way you explain CEA your blog. You wrote:
"By contrast, there will be another group of people who have had—through no fault of their own—the misfortune to get an illness that requires an intervention that's not very cost-effective (surgery, say)."
Exactly. Using CEA analysis both gets more health for the money and allocates resources based on whether someone was "lucky" enough to get a disease or illness with low CEA interventions. This seems fairer to me, even from a common sense perspective, than the explicit lottery approach.
(Small tweak in relation to the parenthetic note: some forms of surgery are extremely cost-effective!)