It's actually pretty hard to hurt yourself on the active blade of a push mower. I've only heard of one account of this happening first hand, someone 1) slipped while 2) gripping and not releasing the handle interlock switch as they slipped while 3) pushing the mower uphill and 4) wearing flip flops. Don't do that, and you'll be fine.
I don't have any data but I'd assume it's vastly safer to mow the lawn than drive a car.
Probably depends heavily on how you normalize the risks, and I had a good chuckle at the thought of “injuries per 100k miles” applied to lawn mowers.
I would guess lawn mowers are higher risk per time or distance, but lower prevalence in injuries per year. I would guess the injuries are mostly a) hit rocks, b) did something dumb with the mower, or c) general outdoors risks like slipping or being bit by snake, etc.