First, Israel's nuclear capability was first developed in the mid-late 60s, when Israel was in considerable existential risk and while the US was not nearly as closely aligned with Israel as it is now. Its development now would most likely result in much more international pushback.
The missiles are deniable, as Israel has never acknowledged its own nuclear force. They serve just as a well as a deterrent without it, so there is no need for official recognition.
The given South Korea example is somewhat besides the point as a) didn't concern nukes b) didn't concern punishment and c) did concern conditions for a weapon/technology transfer which are something entirely different.
Israel has a close relationship with the US (some would say unhealthily close). From an US point of view, rather than being "on the hook" to guarantee Israel's survival in extreme conditions - say another, unlikely, Yom Kippur war, or an Iranian nuclear strike - Israel's nuclear force allows it to carry out its own deterrence. As a long as appearances are maintained by deniability and the likelihood of an Israeli use are extremely low, or could appear justified if it did happen, there is a fair bit of upside for the US to a very close ally being able to defend itself without needing to draw in direct US military intervention, especially of a nuclear nature.
The only real downside is when nuclear non-proliferation is pursued against states like North Korea, where there is a risk of appearing to have double standards. However North Korea's behavior is so extreme that it has few friends anyway and Israeli nukes are not a significant factor there. And, again, deniable. Iran? Somewhat isolated diplomatically, not extremely popular in the neighborhood and if anything, they tend to give quite some bit of justification to Israel's forces.
Is this "fair"? Not something this answer is trying to address.
Note that both Pakistan and India openly "went nuclear" with rather limited long term downsides to both.
Last, the ongoing controversy about Israel's actions and unconditional US support has little to do with those nukes and much more with the failure of Israel to reach a fair agreement with the Palestinians (who, as a comment mentions, are not at all the potential targets of these nukes).