"One of my worse bike crashes was much like that"
This is what I find fascinating.. One of your worst crashes, which implies you have several bad ones and a number of less serious cycling accidents.
My main mode of transportation has been the bike for the last 35 years. The number of accidents I have been a part of during that time is 2 or 3 and those were very minor altercations resulting in no damage or injury and they did not involve cars at all. This is not the result for a lack of trying either. In my... formative years, I have had BMX bikes, mountain'esque bikes, fast sporty bikes and flimsy cheap bikes.
I do recognize that cheap bikes or poorly maintained bikes are more... error prone. In fact, all of my near accident experiences with bikes are due to cheapness or lacking maintenance during operation. There is such a thing as a bad bike. Poorly designed and constructed bikes appear often in countries with no cycling culture. I think it is criminal that, for example in the USA, people are sold tour-bike facsimiles as a mode of transport in cities. That is absolutely nuts!!!
In a busy city, regardless of cycling infrastructure, most, if not all commutes are equally fast or faster on a 3 speed folding bike compared to a 27 speed Giro D'Italia resembling ultra-light racing bike. The only difference is that the folding bike is safer because you are seen better and you can see more yourself.