Update:
Today I did the following. I started the mower up, and poured plenty of seafoam into the air intake. After it had "enough", I smothered it till it stalled out (which was surprisingly difficult!). Then I put about 3/4 of an ounce of sea foam in the oil. Then I put a brand new air filter in there! ...replacing the empty whole where an air filter was supposed to be! I let it soak for a while (while I weed-wacked), started it up, and mowed my whole lawn with it. It's a big yard so it ran for a while. I have also run lots of seafoam through the fuel this year, and the gas it's running on now is brand new gas with a 1oz per gallon dose of seafoam in it. It *may* have made a little improvement, I haven't used this mower enough lately to really be able to know for sure. But it was only after all that, that I left it running for a bit and made the video below:
This engine ran so smooth when it was given to me. It literally ran like it was brand new. I'm bummed if it's permanently damaged from not having an air filter. I had no clue it didn't have one. I'll probably do the lawn again next week, then check the oil and see if it looks particularly dirty, and if so I'll change it again (it's practically new oil, only ~2 hours on it). Maybe another week of use with cleaner in the gas and oil will do the trick, although I suspect it's not going to make a difference.
Any thoughts? Unfortunately my phone never seems to do a good job of capturing the sound of lawn mower engines.