PM,both bishul and any of its related actions are assur on Shabbos, whether kibui, stirring the food on the blech, adding cold to hot food while on the blech, putting raw food that was not cooked before Shabbos ont he blech, etc. All of that is muttar however, on Yom Tov, because they are actions needed for cooking, a meleches avoda which is 100% muttar to do on YT, though not on Shabbos. Also, turning the fire on and off is only muttar on YT when it relates to the cooking. You can’t do it to heat up the room on a cold Sukkos day. Therefore, it would make no sense to say that kibui which is permitted on yt would be permitted likewise on Shabbos, since the permission for kibui on YT ONLY pertains to the cooking which is a permissible action. Since cooking is assur on Shabbos, any related action such as kibui would also have to be assur.
That is my understanding of this, and while it is true I did not learn gemarah and mishneh berura, etc., I grew up in a frum home, and learned from my mother and both grandmothers, just as she learned from hers before her, and my dad’s sisters learned from their mother. If you think about it, all of the family kashrus is really for the most part dependent upon the eidus of the woman who run the household. We did not initially learn it from a sefer, we were taught by hands on mesorah what to do and how to properly do it, as well as what not to do.