Define "London". In inner/central London, where cycling infrastructure is being built and where the Congestion Charge is in place, the cycle commute share is significantly higher than 5%. (Back in 2011, for example, it was 15.4% in Hackney: it will be bigger than that now.) In the outer suburbs, where there's no infrastructure and no Congestion Charge, it isn't above 5%. That should tell you something.

Meanwhile, you talk sneeringly about "invest[ing] billions into poorly designed public transport". Public transport is, of course, something London does very well. Car commuting across the whole of London, inner and outer, is around 25%. Public transport commuting is around 50%. (Source: LTDS. This covers London residents only, so is probably an underestimate given the vast numbers of people from outside London who commute into the city by rail.)

You're right about that. London does public transport much better than most other cities.

It's also been measurably more successful than many other cities in driving cycling adoption. But if your argument in support of that (overall relatively minor) success involves statements such as "define London", and cherry picking particular suburbs, then I think that speaks to the strength of it.

Many suburbs (and inner city London local authorities) have stymied Transport for London efforts to introduce cycle infrastructure. Hardly fair to sneer at cherry picking

Try comparing the numbers of bicycle commuters versus automobile commuters and doing the same for any American city and prepare for your mind to be blown.