GAH! Update: total failure!
The mower behaves the exact same way, rough starts and sputtering/misfiring while running!
I put the new carb in there, with the gasket that the old carb was missing. I also replaced the short little fuel line it has.
After mowing half my lawn with the same poor running, I went and grabbed my primary mower and swapped spark plugs. I hadn't replaced the plug in this machine because it looked like it was in great condition.
So I swapped the plugs and ran both mowers, no change. The honda mower still ran great with the TB's plug. And the TB still ran like poop with the honda's plug.
The gas tank looked totally clean and it was running on brand new gas.
What else is there? What else could cause poor running?
Edit:
here's video of it running. It actually might be a little worse now, at least on startup. Once its running for a minute or two, it's about the same as before.
One thing that's weird, when it starts up cold, it runs extra bad. Almost like it might die out. It stays like that for a minute or so. Then suddenly it changes and goes into what I'll call it's "normal" mode, where it's still misfiring and running rough, but it does so at a much higher and steady RPM. It is totally useable in this normal mode, but it's definitely not running the way it should.
I'm not really sure where to go from here, any suggestions?