This is a lot of why I became a leftist. It was by gradually realizing that for certain goods and services such as health care and education, the government can often offer better and cheaper services. Sometimes incentives in particular types of market become such that they encourage higher prices and worse actual quality.

One has seen examples of this in e.g. Sweden where the privatization of the school system led to a competition where schools cut in resources directed towards improving quality of teaching and direct funds towards things which make the school "look" good, such as flashy laptops, nicer lookig rooms etc.

I suspect similar things have happened at American universities where more and more money has gone into sport stadiums and other things of only peripheral relevance to the quality of teaching.

But I think you cannot entirely get away from ideologial consideration. E.g. if I am a top earner then seen from my selfish perspective it will be cheaper for me to pay for private health care than to pay tax payer funded health care. It is the country as a whole which benefits from tax payer funded health care, not so much the affluent class.

But if I prescribe to the ideology that health care should be a universal right, then I migh favor it still even if I personally don't 'benefit from it.

Although I can see many ways to sell universal health care to people who are only motivated by selfish considerations:

1. You can point out that if you are unlucky and get poor in the future, you are still secured health care.

2. Public health care will avoid the problem that you are seen as a money bag by for-profit health care. One can point to statistics about the US showing massive overtreatment of the affluent. People getting invasive surgery they don't actually need. A demonstration of why health care should not be strongly motivated by a profit incentive.