You are, unfortunately, mistaken. The President has sole authority over launch decisions. There is a requirement that those orders be verified as authentic by the Secretary of Defense, but assuming POTUS gave the orders, it's not within SecDef's authority to countermand them. What would actually happen if an insane POTUS were to give a launch order that made no sense is probably up to chance and the quality of those willing to break the law and refuse the order.

https://en.m.wikipedia.org/wiki/Nuclear_football

"However, the President's authority as Commander-in-Chief is not unlimited and US law dictates that the attack must be lawful and that military officers are required to refuse to execute unlawful orders, such as those that violate the Laws of Armed Conflict."

Those laws are very very vague and like you said, it's possible someone in the chain of command could argue it's against the law since a nuke would likely cause collateral damage one way or another. I think it's technically incorrect to definitively say that the US president has complete and total control over nuking a target.

Hey, wasn't this the EXACT argument Democrats made about removing Trump from office before the election? Floating the idea he'd risk WW3 instead of losing an election?

Turns out Trump's general hard line on everything was useful for something.

the President can order nukes to launch. there's no law curtailing that power, and military orders are the purview of the Executive under the Constitution, so it's doubtful Congress could even prevent that.

the nuclear football is basically a fancy sat phone though, so it's less clear if the generals who get the war order would actually obey.