the actual criticism by those on the left is that private ownership of the means of production necessitates an inequitable heirarchical relationship between an ownership class and the working class.

I have to strongly disagree here. Human nature results in inequitable hierarchical relationships between upper and lower classes-- not just between the rich and poor, but between the dominant racial / ethnic group and minorities, between men and women, between cis heterosexual people and queer people. No matter what kind of economic system we use, assholes are going to try to game it to cement themselves and their cronies in power, and the out groups they hate at the bottom. It happens in capitalist systems just as often as it happens in communist systems.

The only way to counter it is by building safeguards that prevent people from accumulating too much power, and prevents those who do have power from abusing the power they have. (That's why I'm personally skeptical of anarchism; removing all safeguards that prevent power-hungry people from consolidating power is deeply concerning to me.)

this relationship is enforced by the state, which is essentially an abstraction of capital, by inflicting violence in order to protect the interest of the capital owning class.

True. The state is also the abstraction of minority groups, inflicting "violence" in order to protect minorities from having our rights trampled by the majority (although I'd personally go with "arresting fascist motherfuckers"). It's both at the same time, because in a democracy the state is a giant organization made up of thousands of legislators and literally millions of bureaucrats. Oh, and they're all divided into dozens of factions with completely different goals that are at each other's throats all the damn time. Attributing one set of motives to such a diverse group of people doesn't really ring true to me. Especially when you'd need literally millions of people to be in on this evil plot for it to work.

neoliberalism is conservative in that it functions to conserve this status quo, offering incremental material benefit as a social pressure-release valve, but liberal democracy can not deliver liberation to the working class

When my grandmother was my age, here in the US she couldn't legally open a bank account, divorce her husband, or pursue charges against him if he r*ped her. Jim Crow was a thing, and gay marriage was illegal in all 50 states. Liberal democracy has already delivered liberation to us-- except I don't like that passive voice, we the people fought for a better world, and thanks to millions of people's hard work and sacrifice, we won. There are still huge problems, don't get me wrong, and there's a huge amount of work left to be done. But I can't help but look back at the legacy of my ancestors and feel inspired, energized to take up the fight and keep working for a better world.

most liberals don't consider neoliberalism conservative because the coloqial usage in US contemporary politics is referring to the reactionary position of the christo-fascist right.

Definitely true, but even in Western European countries true Reagan / Thatcher style neoliberals are considered center-right, and "neoliberals" in the modern Democratic party sense are considered center-left. (For what it's worth, the Dems are actually a bit to the left of most major Western European center-left parties on social issues, most notably trans rights!)

Sorry to write such a wall of text, I just feel really passionately about this kind of stuff, lol. Feel free to ignore this post if you don't feel like a political debate ATM (which is totally valid, please don't feel bad if you aren't feeling it).