I would have said that the piston was full of oil from the mower being tilted...ran past intake valve into cylinder.
Did you try and pull the cord with the spark plug out and no compression?
Turn the blade with the spark plug out?
I did try to turn it and it would make a wheezing noise, but stop at a certain point. I'm guessing at somewhere during my futzing, the sliver found its way to a point where the piston couldn't make a full revolution because when I first pulled the mower out, I was able to pull the cord fine, but it just wouldn't start.
Thanks everyone for the help. I've learned a lot. It looks like I need to decide on whether to buy a new engine or new mower. The old mower got at least a decade of use so it had a decent life. Unfortunately, every year something would go wrong where it wouldn't start up on the first outing and that usually meant digging into the carb either a little or a lot. So in some ways, it won't be all that missed as long as the new one isn't worse.