If you allow pedestrians to cross anywhere at any time with right of way, that can work to a certain density. On the other end of the scale, traffic will be at a standstill due to a constant stream of pedestrians. I don't know exactly where on that scale either of those cities is, but the argument that it works in the UK where the density is 5x less seems flawed.

I've posted in the sibling thread r.e. the density (which I believe you have underestimated by several orders of magnitude), however a stroll around london will show you that, excepting arterial roads, cars always have to deal with pedestrians crossing at any time and place, including between you and the car ahead if you come to a stop, or if there's more than a couple meters between you and the next car. Even busy arterial roads will have to deal with people walking across them if there are large gaps or the traffic is slow or stalled.

In practice nobody steps out in front of a moving vehicle (for obvious reasons) and most people cross at the crossings most of the time because that makes sense.

In situations where there are a lot of people at the same time, like say a music festival or a sports game finishing, the police tend to manage the flow of people.

I'm struggling to believe London Brigde area and the square mile get 5x less congested than New York. What areas of London and New York are we comparing?

Greater London compared to Manhattan, apparently. https://en.wikipedia.org/wiki/Greater_London. I don't see what conclusions can be drawn from that comparison, which is why I asked GP to be explicit.