The reason inequality is a problem is very simple.
As inequality increases at scale it means that an increasingly concentrated group has more and more capital. What do they do with that capital? They buy assets, they are basically forced to do so by design.
What happens when they buy assets? They capitalize those assets. An apartment unit now becomes a home others can rent but not purchase.
Rinse and repeat until eventually wealth is so concentrated that the ability for any other individuals to access assets is basically zero. This means those individuals cannot build capital or ultimately wealth and it also means that, even if more resource become readily available, more people cannot afford them. They have to do 2x today what they did yesterday to an achieve the same amount of stability even if their "standard of living" has increased because of a wider swath of goods and services.
Honestly at this point I think that anyone who doesn't see inequality as a major driver of contemporary problems is simply not paying attention to the USA or must not live there. Countries in which it is less of a problem basically only mitigate it by having a state that can provide essentials to effectively prevent the capitaled class from taking them away from people (eg healthcare, as you mention).
Economics is all about the balance of who has access to what resources. We cannot just generate resources and capital out of thin air. One person getting more necessarily means another gets less. Period.