You've mixed a subjective measure "rich country" with an objective one "rich people." I can't think of any situation where people of greater means accept more limitations.
Anyways do you have any examples of "rich countries" that have solved this problem?
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).
Rich countries that have good public transit? Sure, that's easy.
Denmark Singapore Japan Germany Switzerland France UK ...
Here are some middle income countries that have good public transit also:
China Spain Portugal Italy Taiwan
Note that this includes environmental arguments - saying that 8bil people cannot all live like people in Houston may be true but it's basically saying we cannot afford to have nice things.
But the thing is US, and especially major cities, currently can afford them!
Why would "rich people take public transit"? Except for extremely dense areas, driving is faster especially accounting for overhead; goes exactly where one needs, any time; and is way more comfortable. Only those who prefer extremely dense areas and also cannot live close to work/amenities (kind of the point of density) would want it.
