Yes rich people take the Tube/trains in London. It's convenient and often faster than driving, especially at rush hour. Basically everyone uses the Tube except for like, royalty or billionaires or whatever. In fact, people pay a premium (in rent) to live close to Tube stations.
Also, planes are a form of public transport and rich people take them all the time.
This part of your earlier comment confuses me:
> I can't think of any situation where people of greater means accept more limitations.
What "limitations" are you talking about here?