Your cycling fallacy is…
“People frequently break the rules of the road when cycling (ignoring red lights, etc.)”
Much attention is paid to the issue of people cycling through stop signals, whereas the truth is that, regardless of the mode of travel used, some people will break traffic rules. People are no more likely to break traffic laws when they are cycling than when they are driving or walking.
There is also evidence to suggest that intentionally cycling through a red stop light can be safer where the road design is poor, usually because the environment has been designed only with motor vehicles in mind. Thus the most visible form of “red light jumping” by people cycling is when someone sets off before the traffic signals turn green, in order to safely pass through the junction before motor traffic begins moving. The rule-breaking behaviour is often a reaction to a dangerously-designed environment.
Good public infrastructure, designed with cycling in mind as a valid mode of transport, makes rule-breaking less attractive and/or necessary. The bad behaviour of some should not be used as an argument against improving conditions for all.