I've never heard of a mower being too wet to start! Did water get into the intake? The only other thing I can think of would be the magneto shorting out.
I did that once on a Dixon Z-turn! Pulling back on the handles with wheels going full speed in reverse while careening forward down the hillside. I looked like Tom Burlinson in the Man From Snowy River!
we mow in the rain a lot as long as its not causing turf damage. you have to do what you have to do to stay somewhat caught up
I mow if it is light rain and if it is not causing any damage. You can't avoid wet grass in this profession.