if it runs as along as you hold the button down ?
you have lost a signal
either the low oil switch or the power signal to the relay that is connected to the low oil switch
the switch provides the ground for the relay, the OK to run signal aka power comes from field voltage supplied by the control board
the board is the third suspect, some times something goes bad between the relay and the voltage regulator,
you will probably end up replacing the board, old models the voltage regulator was a separate component board, more modern models its all on one board
there is nothing in the carb that is used in running and not used in starting and vise versa, with the exception of the choke when cold, the fuel pump must work be working or you would NOT get repeated starts while trouble shooting
fuel pump failure is pretty easy to diagnose, NO fuel in carb bowl
many a fuel pump has been replaced and did not help, when it was a control board problem
the start function bypasses the low oil switch and OK to run circuit, because there is NO latching signal until it is running
Chris or Doug may come on with a better explantion