The traditional problem in a Briggs and Stratton engine that won't start when hot is that the valves are out of adjustment. As the engine wears the valves sit lower in their seats and they'll leak slightly when the engine is hot because the valve stems expand with heat. An engine so encumbered will run but not start, at least not willingly. Your car doesn't have this difficulty because its valves adjust automatically, but a small four-stroke engine needs an occasional valve adjustment. When I last did this I fashioned a feeler gauge out of a plastic bubble pack and gently applied a file to the valve stems, for the older B&S engines don't have adjustable valves. Yours may be improved in that regard.
Mark Kinsler
Mark Kinsler