I followed this video. Same engine as mine. What happens is that the little plastic post rubs up on the metal bracket and get's stuck. It's on a spring and in the video he pulled it towards him, let it touch the bracket and let go. And it got stuck. So that flap is stuck open when you go to start. So he shaved off the excess plastic on the left side of that post and then demonstrated that it no longer comes close to touching the bracket. I did the same thing and my mower started right away.
Well, the second time I went to go start it, which was a week later, it took a dozen tries. And this happened three days ago as well. Both times, the flap is not stuck open. It's closed like it should be for startup.
And the new filter I put in got gas on it again. But that first time after the repair, when it started right away, there was no gas on the filter. If I remember correctly, I checked it after that initial mow.
Now, as you can see, gas is sitting in there so it's leaking from somewhere. Would that be related to the carb float valve not shutting off the supply like you said?
The guy also has a video for the same mower as mine for a carb cleanout. I suppose I could try that.
Design Flaw:
Carb Clean: