Your cycling fallacy is…
“People frequently break the rules of the road when cycling (ignoring red lights, etc.)”
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.
But there is also some evidence to suggest that where the road design is poor – usually because the environment has been designed only with motor vehicles in mind – intentionally and carefully violating a traffic rule may be safer. For example, 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. Therefore some rule-breaking behaviour could be a reaction to a dangerously-designed environment, although of course this doesn't absolve someone who cycles in a manner hazardous to others.
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 (a point that doesn't even need making for other modes of transport).