Yeah you've lost me a bit there. You just defended these classes as natural and impossible to stop so what do you mean by "deal with"? It's an interesting choice of words by the way.
Yeah, I'd appreciate it if you don't do the "interesting choice of words" business. I'm not insinuating anything.
If it's the case that hierarchies become evident in any society, that's not an immoral thing to observe. As long as you have a society with leaders, or some people who have different kinds of roles, some people who are stronger or more charismatic etc. as long as you have institutions that can leverage power, you can end up with some people able to wield power or influence.
The leftist position tends to say, we can in some way eradicate these kinds of differences and keep them from emerging if we do certain things. The right position says, no we can't do that in anything like a complete way, so we have to manage them by creating institutions that minimise exploitative hierarchies while encouraging positive ones - what that looks like depends a lot on what their politics looks like.
The difference isn't that the evil right somehow thinks that hierarchies are so amazing while the enlightened leftists don't. The real question is whether or not it is correct to think the emergence of social hierarchy can be prevented at all. If it can, then the right POV is potentially standing in the way of getting rid of them. But if it can't, the left POV is likely to propose solutions that cause more problems than they solve and also be a waste of time and energy.
There's often a lot of practical overlap between many of those kinds of right approaches and class based leftist approaches because so far, no left government has actually succeeded in abolishing class or other hierarchies, so often they are working within the same kind of social landscape.
The more natural opposite to both of these types of thinking IMO is the kind of radical individualism that you see in neoliberal economics or social progressivism that doesn't admit to things like social groupings or class at all.