We'd have to have an example of public transit done well to break the caste stigma you referenced. I don't think anywhere in the US is anywhere close to Amsterdam (discussed in video you linked)
Celebrities, politicians, billionaires all ride the subway all the time. New Yorkers know to keep to themselves out of politeness not safety and honestly are more likely to step up and defend someone famous being harassed than join in. We're all just trying to get to where we're going and the subway is almost always the fastest and most convenient way (not to mention cheapest) to do that.
According to you? Riding subway in NYC and you'd see plenty of rich people. Go to any station near the financial district, or Park Ave.