This might compound the public-transport issue to the point of significance, but such a cultural element won’t be eradicated by simply providing more “poor people’s transport”.
That definitely does not tally with my experience in Manchester and Liverpool. Nobody worth any real money lives in the city, they're all in big Cheshire villas.
The young professionals might like to play Friends for a bit, but as soon as they spawn or make real money they're off in a blast.
I'd be surprised if Birmingham were any different, considering that city centre in practice is almost non-existent.
London and Edinburgh are the only places with real elites living in cities, but they are the exception to the rule.