Final update:
I managed to get the flywheel off using a combination of the flywheel puller, a hammer, and a crowbar. The flywheel key was sheared, but I think I originally sheared it in the process of getting it off... I replaced the key and still no luck, it backfires and smokes but that's it. I bought proper feeler gauges and the valve gap was perfect, seemed to have been adjusted recently.
So, after spending around $75 and hours of my time, I've given up. It's not worth it, so I'll be giving it away soon. Today I bought a similar compatible engine for $100, a 12.5HP instead of the original 13HP. Supposedly they're practically identical, just the 13HP's carburetor is tuned a bit different. Both are from around the same year and are in the 28000 series. Almost done swapping the engine, should work fine. The seller ran it in front of me, pretty confident it'll survive a few more years in a mower. Thanks to everyone for the help!