I'm just an enthusiast, not an expert -- buuut...:
Direct impacts of modern nuclear warheads on population centers and key military infrastructure are rather limited in terms of their area of effect; what makes a nuclear war so feared is the nuclear fallout that follows.
As a concept, Burevestnik is essentially a lower-atmosphere, subsonic HGV with indefinite endurance. But what does a nuclear-tipped, nuclear-propelled cruise missile actually bring to the table that you couldn't achieve by simply building more (& better) ICBMs and glide vehicles?
If the idea is to decimate the enemy after they've already spent their interceptors and lost most of their sensor and C2 network, why not simply overwhelm their defenses even more during the 1st strike, or even the 2nd? Established systems are inherently cheaper, and they also happen to be harder to intercept (even if much easier to detect). The only real advantage this cruise missile has for its intended mission is its endurance and ability to repeatedly change its flight path -- LO design matters less than people think within the horizon, and it's ultimately bound by physics how much you can reduce its RCS...
At face value, this is essentially just a delayed warhead delivery method. And while delivering it (before, or) after the fact may increase its chances of success, that doesn't necessarily mean it increases the overall success of your nuclear war...
Additionally, if you want more systems to survive the enemy's retaliation for your 2nd strike, why not simply build more bunkers that can survive the nuclear fallout? The whole point of this weapon is to catch the enemy with their guard down, but any launch and increased activity in the run-up to it will still be picked up by enemy space assets, while they'd also be far more alert during a crisis anyways.
If you launch it after M.A.D. occurs, while you can assure that its individual mission succeeds, you could have decimated the enemy even more and received less retribution in return by increasing the size and complexity of your 1st strike instead...
And this weapon is developed as a negotiation tool, but by the time nuclear fallout occurs, all negotiations would have already failed and neither side would have a functional state structure remaining anyway... And who even would want to rule over a world doomed by nuclear *Fallout* (pun intended!)...?
In the end, there are indeed certain use cases where such a weapon brings value. But frankly, it would make more sense for the US to actually develop such a capability rather than Russia. And it can't really remain a niche capability if you want it to be effective, but it can't become a widespread capability either.