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.
Use of shared infrastructure for all vehicles is risky when the rules of movement (position and right of way) rules are not followed. Some cyclists do not follow those rules and end up in collisions. Other times, authorities paint lines that guide cyclists to ride in unsafe areas (too close to the edge of the roadway, or too close to parked vehicles), or designate areas for cyclists to ride where they're hidden from the motorists' view until both enter the intersection.
When one follows the rules of movement and rides in a predictable manner, that risk is largely eliminated.
> We already have sidewalks in many places for this reason
Sidewalks or side paths that have cyclists follow pedestrian right of way rules on approach and through intersections simply doesn't work. The reason is that cyclists move much faster than a walking pedestrian. Pedestrians walk between 2 to 4 mph, while cyclists ride between 10 to 20 mph. A pedestrian that's within a few feet of entering the roadway can be seen by a motorist in time for the motorist to stop and yield to them. On the other hand, a cyclist can be 50 feet away and not seen by the motorist before the enter the intersection. So, instead of yielding, a collision happens instead.
Cyclists move closer to vehicular speeds as opposed to pedestrian speeds (you can't ride in a straight line when going at walking pace). It makes sense for them to be treated like vehicles and follow the same rules. The rules are designed in a way to accommodate vehicles moving at different speeds.
More seriously, you can easily hit a rock or something and go over the handlebars on a bike path.
So, yes, physics.
We should also be demanding that there is no crime as well.
But outside of this fantasy land you need to accept that cyclists will be interacting with cars at multiple points in their journey from A to B. There are just too many practical issues building an entirely seperate cycling network.
So until this magical day cycling should be considered risky.
Just throwing your hands up and not asking for change is a sure way to not improve things.