PM – I think it is clear that oomis will not allow herself to be convinced that she is mistaken, but it is still worthwhile for you to make it clear to everyone else that what she is saying goes keigen halacha. Even with an old pilot lit stove.
P.S. IIRC, the psak that allows one to lower an existing flame is almost certainly NOT R’ Moshe zt”l as nishtgeshtoigen suggested. I’m pretty sure he paskened that you must light a second flame instead. But a few other poskim did allow lowering a flame. None allow extinguishing one (or relegating the flame back to below the stovetop – an action that if taken for the food could be more easily done by simply removing the pot, thus doing so is not in any way l’tzoirech).