I don’t believe it’s fuel related. As spark disappears after warm or hot
I tested the coil with meter both cold and hot and had same readings. Coil was actually cold to touch when it shut down.A common problem with tar-packed coils is they warm up and internally short much more frequently than oil-packed ignition coils (wow, now you know I'm old school if I know that, smile).
That said, this thing doesn't have a pickup coil or electronic ignition, does it? They too can work fine when room temperature but then have an open when hot and fail.
You should look. If the sensor fails, it could/does cut off the ignition.Not 100% sure, but oil is good and full.
If it has a warning light, there has to be a sensor/switch. Find it on the motor and unplug (or maybe "short it" if it's got two wires). This will eliminate that possibility...I don’t see an oil sensor on the wiring diagram. Only an oil light.