Wasn't the standard model for how a nuclear war would start basically "if you escalate, the reasonable expectation is that the other side will escalate as well"? If you escalate despite the nukes on the other side, it would be risky to assume that the other side is unwilling to do the same.
If you escalate, you signal your intent to start a nuclear war if the other side does not back down. But if you are willing to do that, you should assume that the other side is equally willing. Otherwise why would they waste money on nuclear weapons if they are not prepared to use them?
Much of the cold war was basically figuring out what you are allowed to do in a proxy war, so that the other side still interprets your actions as a legitimate proxy war rather than escalation.
Inasmuch as I've read, and that's mostly 60s/70s stuff, most stable escalation theory is based on messaging and red lines.
I.e. identifying a future possibility and loudly proclaiming that if (and only if) it happens, then this nuclear action will take place (and stop there)
Essentially, trying to avoid surprise and uncertainty by tying specific actions to reactions, with the hope of avoiding runaway escalation-begats-escalation scenarios.
Which is notably the opposite of what Russia has been doing, which is more along the lines of "If the West pushes us too far... they're in for it", which is more the kind of bluff you'd make if you never intended to actually escalate (because you create the fear without establishing any requirements to escalate on yourself).
While the Cold War was still raging, Thomas Schelling, of 'Strategy of Conflict' fame, pointed out that while the war stays cold thanks to the 'mutually assured destruction' consensus, it would be an optimal strategy in terms of bang-for-your-buck to pretend to have many more nukes than you actually have.
Concretely, since no one ever wants to call your bluff and test your nuclear arsenal, it's natural to expect that unless there are strong countervailing pressures, one can expect that many nukes will be badly maintained and turn out to be duds.
Given what we learned about the state of the Russian armed forces since February 2022, it would not surprise me, if Russia barely had any working nuclear capabilities in practice.
(However they would still be dangerous enough. Even a nuclear payload that doesn't explode properly would still be a very dirty bomb.)
Without wanting to do any speculation how the current situation will ever end, but I always assumed that the first stage of nuclear escalation would be weapons tests - to exactly call this kind of bluff of the other side and to signal that the own side would be actually capable of performing a nuclear strike.
From what I know this was also what happened in the cold war - the nuclear threat back then was a lot more "palpable" than it is now, possibly because both blocks were already constantly launching nukes - they just weren't launching nukes at each other:
Yes. Though the test would only tell you that the other side has at least a few working nukes, not the proportion of working nukes.
(If a side could credibly commit to testing an unbiased (pseudo-) random sample of nukes, they would be able to convince the world that a large portion of their nukes work.
But I am not sure how you would do that without revealing all of your nukes?)
Spies exist. You never are sure how many the enemy has or what they are leaking - even if you do learn, that is a snapshot in time: tomorrow they may get another spy and then you are back. As such you are risking the enemy learns the nukes don't exist and then the whole thing fails.
We now have treaties to inspect each other's nukes. So unlike other things we have a good idea of what how many exist and are working - or at least a floor on the number (that is either side may decide to risk the treaty with secret nukes elsewhere. Between the potential of the program being discovered and the costs I doubt it, but I must allow this for completeness). As such it is generally believed that Russia's Nukes are in mostly in good shape - those who should know all seem to be acting like they are.
Official inspections can peek and poke the nukes a bit. But they can't really tell exactly how good the maintenance is. Similarly for a spy.
You are right that wholly non-existent nukes would be harder to fake. But covering up bad maintenance isn't quite so hard.
In your point's favour: Looking at the Russian military budget, they do seem to spend a lot on those nukes. Of course, funds could still be siphoned off. But having a big budget is at least necessary, even if not sufficient.
If you escalate, you signal your intent to start a nuclear war if the other side does not back down. But if you are willing to do that, you should assume that the other side is equally willing. Otherwise why would they waste money on nuclear weapons if they are not prepared to use them?
Much of the cold war was basically figuring out what you are allowed to do in a proxy war, so that the other side still interprets your actions as a legitimate proxy war rather than escalation.