This wouldn’t be capitalism governing itself in a neoclassical manner since central authorities would be instrumental in their role as the demagogues that control the governmental (or corporate, monopolistic) factions.

Agreed, if that's how one views the government. Either the government is "they," or it is "us." But, of course, such dichotomies are always reductive. I'm just saying that to the extent the government constrains capitalists, it is acting as "the market," i.e., as part of the self-regulating system. My question is whether the great minds behind neoclassical economics would have changed their notion of "self-regulation" if they had the benefit of our hindsight. They had only seen the governments they had seen.

I have in mind Keynes's advice that governments run surpluses in good times and deficits in bad times. When he was working, the supply curve was a fairly sticky thing, so government's role was to steady demand for the relatively fixed supply of goods. In the time since Keynes, however, production/delivery methods (including ocean transport) and product mixes have changed so that the marginal cost of many goods remains low even as demand rises. (I'm speaking ex-COVID, which has really effed up the supply chain and will do so until the West de-Sinofies for strategic reasons.) Given low marginal costs, I believe that Keynes would have recommended running deficits in good times and bigger deficits in bad times, growing demand to meet potential supply. Same mind, different facts.

Potential supply is a technological artifact. So, I submit, are collective bargaining and the kind of politically negotiated boycotts (laws) now available to the capitalist system. I doubt that Smith and Mill et al. saw what would become possible. If they did, I wonder what they would have said.