I always run my mower till it dies (out of gas) on the last mow of the season and never had a problem with doing this on a B&S...but it doesn't seem to work well on the Honda unit.
I do the same thing each fall/winter, add fresh gas in the spring, remove the air filter, and start it. It always starts with 2-3 pulls. After that, during mowing season, it always starts with 1 pull, occasionally it takes 2...
I definitely wish mine had the 190cc engine at times, especially in spring when the grass grows uncontrollably and I can only mow 1x week. It can bog it down in the thick, tall grass.
I have a similar Honda powered mower. The gas cap won't let the tank breathe, and after a few minutes the engine dies. Took me a while to figure that one out. The parts inside the gas cap must have swollen.
The serial number is to pull up the correct drawing. I wanted to see if the carburetor has any mixture needles and I wanted to see how the fuel system is vented. I suspected you have/had a vent problem.
You have a faulty carb or a plugged fuel cap vent. When the hot engine expands the fuel it forces too much pressure (more than gravity) on the needle and seat and it will flood over.
I would still get a new carb or inspect and check the float.