Even while Rav Moshe’s s’varah for allowing gas to be turned OFF totally on Yom Tov (and thereby the flame is gone), should also apply to Shabbos; he was never matir to do it on Shabbos.
In addition, we have no Rav Moshe today in the global sense that if he said the Kosher-Switch was permitted, Poskim/Rabbonim would pay attention.
It wasn’t that long ago when recognized Poskim mistakenly paskined that using electric appliances on Yom Tov was allowed (fire-aish was in the wires they believed) and indeed large populations of yidden (including Chasidim) followed this Psak. So here we have a precedent for Yom Tov use of the Kosher Switch for lights and appliances.