My impression is that, while cycling might be more dangerous than driving on a per-mile basis (ignoring health and financial arguments), the general perception of danger seems far greater than reality. Most of the time I mention biking in greater Los Angeles, I get incredulous looks and responses, "Isn't that dangerous?" I shared that perception before I started doing it more frequently, and the first few times I rode on higher-traffic roads, I was quite uncomfortable. However, I learned quickly with experience; a lot of this is in learning how to treat cars/drivers, asserting yourself when need be, being very visible, following the laws, wearing proper gear, assuming that all drivers are distracted, etc.
Being a pedestrian is actually extremely dangerous here, by any measure (far more pedestrian traffic deaths than bicycling). Why don't people get aghast at the danger when you say you're going for a quick walk? Distracted walking is a thing; distracted cycling (with few exceptions) isn't.