The big difference between the two camps that I see is that one side recognizes the cost benefit of their position and the other side doesn't.
Price controls on healthcare (socialization) has positive and negative impacts. The positive impact is universality. Everyone gets approximately the same basic level of care. In some countries you might be able to pay for a nicer room, or quicker service, but generally speaking it's all the same. The downside is that price controls stifle innovation. That's just basic mainstream economics and is the case in every industry, not just healthcare. The trick about stifling innovation is you don't feel it immediately. On day 1 of the price controls, the rich are about the same but the poor are better off. On year 50 of the price controls, everyone is worse off because of all of the innovation that never came to fruition.
I never hear advocates of universal healthcare have an adult conversation that they are trading long term benefits for short term gains. I think that's a reasonable position to hold, but if we just pretend that it's all benefits and no cost we're not having an adult discussion. It's not as if one side is right and one side is wrong. Once we accurately discuss the economics, it's easier to understand that this is a very difficult moral decision and not the trivial one that is sometimes portrayed in the political sphere.
My speculation is that the rich people you refer to are more likely to be trained in economics (formally or otherwise) and are more likely to have a long term mindset regarding cost benefit tradeoffs.