Good public transport, in a city at least, doesn't feel like a limitation! It's rather convenient, really. There is no problem with making rich people use it, they do it voluntarily.
I think it does genuinely feel like a limitation in a lot of cases, and I think it's unhelpful to pretend it doesn't. However it also is very convenient in others.
Everything has it's place, in an ideal area if I was wanting to go a short-medium distance I'd go on a bike, if I wanted a drink or to carry more than practical on my bike or to go a medium length I'd take public transport, and if I wanted to take a lot of stuff, or go in a time sensitive manner, or not want to worry about the last bus, or go somewhere rural I'd take the car.
I don't think good public transport can replace cars in all situations, but it is an extremely good option to have for situations that suit it (which for most people is likely most, but probably not all, of the time).
No, not absolutely all situations. But take my situation - I live in Berlin and I use my car less than once a month. A major reason to drive is to have a car at the destination. I've also had streaks where I drove to the supermarket every couple of weeks, but I haven't done so in months. Main upside is that I can buy whole cases of beer and a bunch of other drinks and heavy stuff, less trouble carrying. Otherwise, the closest supermarkets are about five minutes by foot.