When cycling for sport you should always wear a helmet.
When I spent time in a French hospital after bike helmet use (not involving a car by the way, except for the ambulance that called the helicopter), I was really curious if I would continue that pattern or become of of those "helmet even on civilian clothes rides" people. Was expecting the latter, but nope, would still feel as alien as putting on a helmet to drive.
Now you can decide you want to bike faster, and by all means (and I would make this choice as well), when riding aggressively in any context, you should wear a helmet.
Mutual yielding (where two pedestrians approach each other on a sidewalk) works perfectly fine at walking speed. It doesn't work at vehicular speeds, which is why the rules of the road exist that determine positioning and right of way. In order to travel at faster speeds, one must follow a set of rules. Relying on mutual yielding results in the collisions you mention.