It is easy to pigeon-hole because conservative writers were writing about this kind of thing before Marx (about five decades before in fact, this is a slightly complex point because politics then wasn't so linear but a lot of the critique of capitalism came from the wealthiest, it was the new rich Liberal factory owners that were the advocates for capitalism, the traditional land-owning elites, for example Disraeli, were making identical points to the OP).
I think the problem today is that people view Marxism within the context of a discussion that postdates Marx by one century. Historical materialism is also something quite distinct from Marxism. Marx wasn't a Marxist (or so Engels said), and there are lots of historical materialist historians (particularly economic historians) who aren't Marxist either.
On your final paragraph, I think much of the discussion today is totally reductive. You are right there is no difference between neoliberalism and socialism but that is because there is no difference between neoliberals and socialists. I am not actually opposed to "Third Way" politics but there is no real representation for working people. And yes, part of that is because the left has the moral authority and wealthy people have hijacked that moral authority in order to fight their culture wars about social causes that are largely irrelevant to most working people (the enemy doesn't exist, it is just a way for wealthy people to shed their guilt and obtain self-justification). It is telling that working class reaction in the US was led by an orange billionaire, it makes no sense (and btw, I don't think people like Sanders or Warren are particularly compelling or effective leaders, the reason orange man won was because, for all his grievous faults, he actually seemed like a normal human being...as weird as that sounds when talking about someone who is so obviously abnormal in many ways).