A minor addition to your hypothesis is that most cars in London are manual (stick shift) - even if they could find somewhere to park, pay the congestion charge, etc they might well be less able to operate the vehicle too.
Manual cars are more popular but automatics are still easy enough to find. Also if you're registered disabled you may qualify for a specially adapted Motability car.
I was assuming that most people are on crutches temporarily, as I've only ever known people to use them this way. I would assume this group are less likely to get a special car just for the duration or their use.
A quick google didn't give me any idea of what proportion of people use them permanently/long-term, although there are reports of complications increasing with prolonged (>1 year) use [0] so it implies they definitely are around.
