Nah, conventional military is still relevant even in counterbalance to other nuclear powers.
It allows exertion of force upon targets which are not important enough for the other nuclear state to burn for.
Let me put it this way, in the hypothetical situation that the ROC(for whatever reason) were to happen to elect a government with the party line of full and permanent independence from the PRC: do you think that a conventional-arms-poor United States would be willing to burn for Formosa if the PRC decided to push for it? No, I think the US government would decide that California and Massachusetts are more important, much less everything else in between, screw defensive obligations. I'm not entirely certain that the conventional-arms-rich US would decide to pay the price required to stand up to open confrontation with the PRC. But the fact that it could changes the equation. If it thinks it can relatively easily, that changes it more.
It's the same reason France decided that it needed to possess its own nuclear weaponry during the Cold War despite US obligations to retaliate on its behalf in the face of conventional arms inferiority on the European continent. It had decided, quite probably correctly, that the US very well might decide at the end of the day that France and all of Europe combined is probably also not worth Nebraska in a MAD situation. They wanted to possess a trigger capable of forcing the issue.
FB has it precisely. I'd only add that there's some things - reconnaissance, search and rescue, maritime law enforcement, disaster relief and so on - that you simply can't do with a nuclear weapon. War isn't all of the military's business.
I agree with what you both say, but those are other phases in a war and not an 'arms (in this case conventional) decide who will actually win'. Even if one of the nuclear sides suffered devastating defeat in its conventional arms power, the nukes would still ensure that they would not really be at risk of losing much when peace is signed.
Of course some level of conventional army is needed both in normal circumstance (eg the armies of most euro countries are pretty small) and if a nation wants to project power (something i don't like at all, but it happens anyway). But those won't win a war against a nuclear power. They can alter the degree of mayhem caused if nukes end up being used, but by that time it likely will be apocalypse already.