I have a Kohler V twin on a walk behind mower ( sorry, just don't know the model) which acted the way you describe. It has 2 coils (magnetos). When the engine came to temperature, it warmed the coils sufficiently that the "tired" coil's resistance changed enough to cause it to malfunction and not generate spark power. The engine would continue to run on its other "good" coil but since operating/firing on only one of the two pistons, it had not enough power to mow and travel and would conk out when the blades were engaged. New coil fixed the problem. My 1995 F-350 dump truck with Windsor 305 engine did the same thing and a new coil cured that as well. Keep in mind that as electrical circuits and/or wires heat up, their resistance increases and efficiency / conductivity decreases because the electrical energy converts into heat energy and rapidly increases the coil's temperature and further decreases its conductivity. I am not certain, but I surmise that an out of operating range heated metal object, like a wire bobbin found in an ignition coil, has a lot of electrons zooming around in a random / disorganized manner and an otherwise straight through running current joins the fray since it meets up with the crazy electrons which are resistant to an organized flow. An organized flow is the current on which the spark plug's operation depends. The engine "acts like it's not getting gas" because it is starved for current. It is not sparking sufficiently to sustain useful power generating combustion and then sparks so little as to cause it to shut down. After a period of cooling off, that minute you wrote about, the coil likely cooled off enough to again provide sufficient organized electron flow=ignition current.
Having said all that, the fuel vent failure comment is also a strong contending concept and easier to prove out that changing a coil, cheaper too.