Universal healthcare isn't guaranteed to work well. The point is that it at least can work well and you can see examples of countries where it is working well. I suspect that people would feel very differently about private healthcare if it was their only option rather than an additional option for those with the money to spare. For example, in the UK I am glad to be able to make use of private healthcare for small things when I need to, but I would hate to be in a situation where it was my only option. (And that's not because the NHS in the present moment is a particularly shining example of universal healthcare done right.)