But at the same time, we're moving past deterrence. The USSR fell 26 years ago. There's "deterrence" and there's "we can wipe out the entire planet". It's especially disconcerting when you consider that in most nuclear armed countries, US included, the authority to launch a nuclear strike lies solely with one person.
We are in no way moving past deterrence. SALT, START, ABMT, etc essentially codified it.
And deterrence is predicated upon creating the absolute certainty in your enemy's mind that you will leave their country a smoking, radioactive hellscape if they attack you.
This requires convincing them that the decision will be made, were it to come to that. Not "Oh, but a mid level officer might ignore the command."
In nuclear game theory, the less certain your opponent is of your resolve and capability, the more likely nuclear armageddon is.
I'd imagine it makes it less stable simply from a complexity = narrower bounds of stability standpoint, but fundamentally it doesn't change anything.
That's why nuclear fingerprinting and global monitoring have been so heavily invested in. "Safety in numbers" isn't a defense if the retaliatory strike still hits all the belligerents.
Aside from "no one has large nukes", which was never geo-politically practical once the USSR reached parity w/ the US.