I think the source goes against both your claim and its own headline. Diagram 3.11 shows cycling to be a small part of the makeup.
3.11 appears to cover the whole day rather than just the rush hour, and again, the context of AmericanChopper's comments was "commuters". 3.5 in the original report (search for 'Traffic in the City 2018') gives more context as to how modal share varies throughout the day.
Sounds like people didn't want to cycle to work, but it was the next best thing after the car reduction measures.
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.)
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.
(Source: https://www.bikebiz.com/news/traffic-in-the-city-2018 )