Agree with you, air gap and check flywheel key. Just little bit off key will cause such.
When checking ignition spark I like to use the Briggs 19368 and I can leave it inline while engine is running and check spark and also compare the run spark to when one don't start and when they are running. (watch the spark close in a shaded non bright area)
If that is 12v I would probably install a bat for long enough to get through winter so as not having to manual crank in cold weather.
Noticed that appears to be a Horiz shaft about 6-9 hp OHV.
I've had similar Briggs starting issues with other models.
One some if
they started little easier for awhile after a new plug when hot I could throw a $12 clone magneto on some and they would go. The flooding seems to indicate either spark or timing or low compression..
Had a really strange one awhile back. (and a pleasant surprise) A Briggs twin 18hp L head with 12v starter was acting like yours and I had been all over the engine, another mag, checking valve lash, compression, leak down while hot, another carb and new plugs would help for little while but could put the removed plugs in another engine and they run ok and the aluminum flywheel starter rings went bad, couple teeth broke on the aluminum ring and was chewing up the starter bendix. I had a heavy duty bat on it so as I could eventually get it started. It was worse after I attempted a hot re-start 5 min or so after it was killed. I suspect this is the reason that I ended up with this give me engine, but I never look a gift horse in the mouth AT FIRST.
This was about 6 mo's ago I throwed another old flywheel on it that had a good starter ring and it now starts as soon as I hit the key hot or cold engine, starts so fast have to be ready to turn the key loose.???????????????????????(just stumbled onto that one) and it had a good flywheel key.??????
Them mag's have some electronics buried inside and some of the electronics are related to timing. I've found this out by trying to change just the electronic mag coils on some 2 cycle engines. Just because I see spark does not get me a run.