No, that particular sort of Two-Stage Marx is one of the most pervasive misreading's. Though Marx and Engels clearly argue that the tech advances necessary for socialism is provided by Capitalism, and the destroyers of the capitalist class is the working class born from the development of capital, there is no two stage formula about highly developing capital in some controlled way. Check out Radhika Desai's work on how Marx's purpose was not to make a stagist argument. I have a recording of talk she did, not sure if the link is still active but I can DM it.

Also what you describe is not the "withering away" of the state. Beyond dealing with whether Lenin's argument is correct (I would partially challenge that it is, it's mixing up Marx's lower stage of communism with some other stuff), and whether Engels coining of the term in the Anti-Duhring is coherent with Marx, what China is doing is definitely not that. I would argue the wage form and the market, private property, the army, the bureaucracy are decidedly already abolished with the establishment of the DotP if we take Marx's position. I mean Marx is very clear in his writing on the commune that the binding of government to capital must be completely severed for there to be a DotP and for the state (at which point the govt is no longer really the state) to go away. I mean Marx says that self government of the workers would be nothing but "a sham and a snare" if the the workshops did not or were not able abolish the market and freely democratically decide production among themselves.

Also one last note on the objective factors argument from earlier, there's a very good quote from Marx's Inferno that deals with it well and so I'll post it here: "The material conditions of socialism are not the objective factors of industrial technology ... but the proletariat’s felt need for large-scale, cooperative production, coordinated on a national or global scale. This can only be a felt need when capitalist development has broken down the laborers’ reserve of individual skills, so as to make their material interdependence obvious and robust, and when the power of the capitalist state has developed to the point where the futility of worker separatism has become equally obvious. Both of these developments have an objective, technological component. Industrial technology helps to realize the first condition; military and bureaucratic technology help to realize the second. But what makes these conditions material to the foundation of socialism is their apprehension by the laboring classes. The material conditions of socialism are the conditions that matter for its feasibility, and these are, for Marx, primarily the motivational—hence “subjective”—conditions of the mass of laborers. Insofar as these motivational conditions have objective, technological preconditions, the link between the two is not so problematic as in Cohen’s construal. Cohen’s subjective conditions are moral and other-regarding, whereas Marx’s are prudential, and, while there is no reason to think that the level of industrial development has any straightforward repercussions for people’s moral commitments, it would be very odd not to think that the level of industrial development has direct and specifiable repercussions for people’s prudential strategies."

Developing the felt need for cooperation is what births socialism, not developing capitalism in any controlled manner or the state improving working conditions (though this is obviously a good thing), and any attempt to do so on the basis that that develops socialism misses the point. For example the independent outbursts of labor struggle in China we see that the state suppress is in fact the very thing that needs to be developed.