To be honest, I didn’t read the whole back and forth here, but I think an important aspect of mehadrin vs. mehadrim min hamehadrin was not addressed.
Tosfos says clearly that the basic mitzva is one candle per household. Mehadrin is each person lights 1 candle. So if there are 6 people in the house, all 8 nights 6 candles are lit. Mehadrin min HaMehadrin is that on the first night one, second two candles, third three etc. That is that ONE person in the family lights ONE menorah for the whole family. Tosfos explains that if each person lights more than it is not clear from looking at the window what night of chanukah it is.
The Rambam argues on Tosfos and clealy states that if there are 10 people in the house, the father lights 10 candles on the first night, 20 on the second, 30, 40,50 … The achroinim have an explenation for this machlokes. So from Tosfos, a son OR a daughter would not light and the father would light for EVERYONE ONE menorah and the RAMBAM holds that sons and daughters don’t light rather the father lights for them.
It is possible that Tosfos is only refering to where people don’t use a menorah, rather they use class cups standing alone, (a common thing in Eretz Yisroel where people light outside), and one wouldn’t be able to tell which night of Chanukah it is. But if each person has his own menorah (as is the standard in the USA) maybe Tosfos would agree that each person light his own.