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.
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.
[0] https://www.cege.ucl.ac.uk/arg/Lists/ARGnotes/Attachments/6/...