My generator won’t start

Bottom line: your float isn't doing its job. Perhaps it's cracked, and thus doesn't.... float.

As the float comes up, it's supposed to close the needle valve. And you've shown that if you manually manipulate the float, the needle valve is working. So your float isn't floating.

Time for a new float, I'd surmise.

I would replace the spark plug and remove the air filter to make sure it isn't plugged up. Then, while the air filter is still removed, spray a little starting fluid into the carb through the hole under the air filter and start it up. If it runs for just a couple seconds and dies, that means the gas in your carb was probably bad. Starting it with the starting fluid a couple of times will work that out, and it should start running fine. Keep your face and other body parts away from the hole you spray the starting fluid into so you don't get burned by a backfire. Put the air filter back on after you get it running.

Other common fixes are pinched or clogged fuel lines or a clogged fuel filter, but if you're smelling gas, that probably isn't the problem.

You should also check that the spark arrester in the exhaust pipe isn't clogged up with carbon. Undo small bolt adjacent to exhaust pipe and remove insert. All small holes should be clear.