Yes, but there's a key difference with respect to the death penalty [...]

All of those are, however, moral statements, not empirical ones. By now you know which I agree with and which I don't, so I won't touch on that. But while they can make cogent arguments against the death penalty, whether or not it is a deterrent doesn't depend on them.

None of that can be said of laws against sexual contact with children, which is a) overwhelmingly supported by society, and b) not self-undermining.

Really?

The state locks people up for many years, up to the whole of their natural lives, in subhuman conditions, and subjects them to frequent rape [no group is more at risk of rape than inmates, did you know that]? What different thing did the kidnapper/rapist of Jaycee Dugard, other than not having a badge? That fits your criteria for self-undermining.

a) overwhelmingly supported by society

1. Ad Populum.
1.1 as any Ad Populum, if I looked hard enough I could probably find a community which approved death penalty for child sex crimes...
2. Still doesn't say anything about deterrence. Plenty of measures supported by society, not even all of them new crimes or new penalties, are proven not to be deterrents.

Well, it IS unlike every other act of lawbreaking, because ALL acts of lawbreaking are different from each other.

Why stop at "child sex crimes" then? Why not "child sex crimes committed in winter?" Why not "child sex crimes committed in winter against a child of different race than the perp?" If you want to go for specificity, you can always get more specific.

But really, all acts of lawbreaking have striking commonalities. The first one, and a really important one, is that all point out to some guy not abiding by overarching rules of their society. All crimes are, apart from anything else they are, anti-society acts. And beyond that, but partly arising from that, all crimes can be studied with a similar set of tools, anthropological, sociological, statistical, and economical. They are not identical, of course, but they are similar in many respects.

And one of the respects in which they are most similar, is that any time the law bans something, there are only a limited set of answers to that prohibition: (0) stop doing the act; (1) hide doing the act; (2) conceal your identity when doing the act; (3) run and hide after doing the act; (4) shoot the police when they come for you; (5) corrupt the police, DA or judge when they got you. As different as every act of lawbreaking is, all share this set of options.

Which is at the root why penalties do not deter. The only thing which seems to be a general deterrent to crime and unequivocally not cultural [by culture here I mean things like how the East Asian countries are particularly safe and it seems to be a cultural feature] is the quality of the justice system...

Meanwhile, the commonality of the possible reactions overrides the differences - at least talking about deterrence. If any of the options 1-5 is better than option 0, there is no deterrence. Deterrence depends on option 0 being superior than all of options 1-5.

Since the passing of a law banning an act; or the increase of a penalty for an act already forbidden, doesn't change the ease of doing 1-5 or their chance of success, its effect on deterrence is at best indirect, through making the chance that all 1-5 fail costlier.

Hence, the reason they all have different punishments.

Nah, that is mostly cultural, and dependent on what is perceived as worse vs. not so bad; and as acceptable penalty vs. too uncivilized.

However, in order to counter your point about the death penalty, it needs only be different in one key respect, which I have elucidated above.

All moral arguments. Not factual arguments about deterrence. If the police only catches 1% of lawbreakers, or can be corrupted by $2000, then you can write down any penalty in the books, 99% of people will act like they are not part of the 1% which will be caught, and anyone who can pay $2000 at a moment's notice will not even bother...