Next step I would do is pull out the plug, drop a little gas down the hole, put the plug back in and see if it fires. That will take plug, stuck valve and timing out of the picture. Report back with results.
Thanks Phil! I put a few drops of gas in the plug hole with an eyedropper, replaced the plug, and it started immediately. Shut it down and started it a few times, then mowed about an acre without issue.
Thanks so much, everyone, for the help. We had a bunch of kids over today playing in the grass and that wouldn't have been possible if I didn't get this mower started.