No, it’s really not the same thing. You can legislate better schools with a variety of methods, the main point being that you’re regulating government jobs(to oversimplify). You’re more limited to negative legislation for parents, such as punishing child abuse. I guess you could technically legislate certain mandates for parents to be better parents, but like, good luck passing said legislation. And even if you do(and this is the big boi), how the fuck do you enforce that??? And on top of even that, how can you be sure parents will be qualified/able to teach their kids such a wide variety of skills? You can fire teachers for incompetence and publicly investigate school districts for failing to faithfully implement good practice. And it should also be mentioned that shifting these expectations (especially via legislation) onto parents will disproportionately burden the poor who will be less likely to have the time, skills, or knowledge to teach said things.
Eh, I dunno that I’d actually characterize him as a liberal so much as him being an authoritarian that just pushed whatever happened to serve him at any point. Kinda in the same vein of fascists not having any economic ideology, just whatever serves their ideal of the state at any given moment. So yeah, I certainly agree with your sentiment that Stalin certainly was not a communist, but more because he only cared about gaining/maintaining power rather than actually subscribing to any economic theory.