I’m pretty skeptical that switching a car commute to a bike commute increases safety.
Anecdotally I have seen a lot more bad bike accidents than car accidents, despite the number of bikes being much, much lower. This makes sense, since you are much more exposed when on a bicycle and while you are moving slower the cars aren’t.
If the stats do say this (I assume you have indeed read that somewhere), I suspect there is some large selection bias at play. Like, people with a safe bike route available are more likely to bike, which skews the numbers towards safe even though I imagine switching any random car commute to biking on the road would make it more dangerous on average.
Public transit of course makes sense, and I do wish biking in urban areas was safer since I don’t do it now because it’s dangerous and it would be great to be able to do it safely.