> It is that the US has the death penalty for this, and they want to kill people who commit war crimes.

> At least that was the reason I was given in the US military.

That's obviously nonsense. For many reasons:

(1) ICC jursidiction does not supercede national jurisdiction that is actually exercised—that's explicitly a basis for ICC jurisdiction not to be applied (it can be applied in the case of sham proceedings designed to provide cover). So it wouldn't stop the US from trying, and applying the death penalty to, any war criminals.

(2) The actual reasons for the the US opposition, including Congress passing a law threatening the ICC, are matters of public record, and are much more about the US wanting impunity for accused (American) war criminals than any fear of inadequate punishment.

(3) The US response to its own war criminals that it has had the opportunity to punish since the establishment of, and its refusal to join, the ICC has shown a singular lack of capital punishment. And even the occasional Presidential pardon after conviction.

Even for US military propaganda directed at is own personnel, that's pretty lazy, low-effort stuff.