Heads of state are generally pretty good at delegating the C&C of their nukes to people they are pretty popular with. That's orthogonal to popularity polls of the populace.
You could automate the process or compartmentalize it enough so that no one knows they are essentially committing suicide. But in that case you are removing human reason from the loop and your system will be too sensitive.
Essentially you have an automated deadman’s switch. Either you tune it to be too sensitive and the thing goes off because you went out of contact for a few hours—likely resulting in your own death.
Or you tune it to be not sensitive enough and your attacker takes advantage of the delay to take control of or destroy the system.
Why care whomever is in power next? You could just do your job.
So, the solution is to press the nuclear button, get a couple hundred million dollars from an offshore account in Cyprus, and live in any country of your choice. Why care about polls in this hole, and what the US will do with this hole in response to the use of nuclear weapons?
The chances of all of those people escaping the country after nuking the US is close to zero. The entire country would mostly likely be completely destroyed before you could make it out. Even if you did make it out, your friends and extended family definitely won’t.
And good luck spending that money when the US is intent on hunting you down. In this scenarios your boss wasn’t safe with a nuclear deterrent, you’re definitely not.
Why would it?
1. "Nuclear capability" is not binary. The available delivery mechanisms and the defensive capabilities of your adversary matter a lot.
2. MAD constrains both sides. It's unlikely that an unpopular Head of State getting kidnapped would warrant a nuclear first strike especially against a country like (Trump's) America, which would not hesitate to glass your whole country in response.
3. It's extremely risky to "try" a nuke, because even if it's shot down, does it mean your enemy treats it as a nuclear strike and responds as if it had landed? That's a very different equation from conventional missiles. E.g. Iran sends barrages of missiles because they expect most of them to be shot down. It's probably not calculating a scenario where all of them land and Israel now wants like-for-like revenge.