I'm not convinced. If we try to identify the common general principle that HCH arguments are appealing to, it would appear to be something along the lines of:
(P) people ought not to encourage actions that they would not be willing to perform themselves.
But surely (P) is false. I would not want to perform open-heart surgery, or an abortion, or any other medical procedure for that matter. But I'm glad that other people are more skilled and less squeamish about such matters. Am I somehow a "hypocrite" for approving of modern medicine? That doesn't seem very plausible.
So this general principle is false. Is there any other grounding for HCH arguments? It's not necessarily inconsistent to think that something ought to be done, whilst being averse to doing it yourself. But it might involve a sort of free-riding. I think that's what's involved in the war argument, especially. For example, Looking In NZ complains:
The Left is compassionate with other people's money. The Right is patriotic with other people's lives.
The underlying principle here is that one ought not to offload the costs of one's preferences onto others. If you want the benefits of X, you should also be willing to pay the costs - don't expect others to do it for you. (I think this individualistic argument has limited applicability. It is legitimate to demand that others join you in contributing towards a morally-required common goal. Leftists are generally quite happy to pay higher taxes along with everyone else. They are not trying to be free-riders. Quite the opposite: they want everybody to pull their weight, and contribute appropriately towards the demands of social justice.)
In any case, the free-rider argument is distinct from the hands-clean one, and doesn't seem to apply at all to other cases such as butchery or capital punishment. ("You want the benefit of tasty meat, but you selfishly offload all the costs onto those poor butchers!" Doesn't quite work does it.)
In those other cases, I think the HCH argument serves as a rhetorical ploy aimed to engage one's imaginative sympathies. It doesn't really serve as an argument in itself. Rather, it is a plea to think through the real implications of your position. The point is simply to combat ignorance that arises from intellectual 'distance'. It's like saying: "Sure, you can support the death penalty from the comfort of your armchair, where you can ignore or abstract away the details - and even the death itself. But could you continue to support it after confronting the harsh reality?" If you can truly answer "yes", then that's that. The argument has no further force against you.
This "ignorance argument" is quite legitimate, I think. But it's also very weak, as the above makes clear. The literal HCH argument is an illegitimate extension of it. It makes much stronger demands (you don't just have to confront the reality of X, you must take part in it!), demands which are harder to meet, and so have greater rhetorical strength. But they're also entirely unreasonable demands. Because principle (P) is false, the mere fact that I am personally unwilling to X does not exert any rational force on me to universally denounce X (or even refrain from encouraging others to do X). Further, the charge of "hypocrisy" is smuggled in from an entirely separate argument - the "free rider" one discussed earlier - and has no basis in the present context, where the fundamental problem is supposed to be imaginative ignorance, not selfishness.
So, I conclude that HCH arguments lack rational force. Their proponents probably should be making "free-rider" or "ignorance" arguments instead. But if so, they should be clear that that's what they're doing. Avoid the rhetorical questions enquiring as to whether others are willing to get their hands dirty. It simply isn't relevant.