"Mother Flame" seems too related to the "eternal flame" concept (an eternal power source) which isn't only in one piece, and also seems to follow the same naming pattern as a "mother dough".
I think it's likely a power source, that is either being directly weaponized (similar to how a nuclear reactor, our closest technology to an "eternal flame" could be weaponized), or is being used to power some advanced weaponry (Uranus?).
I checked One Piece Red dialogue again and the Elder know that Shanks is the father of Uta. But they only assume that Uta might be a Figarland. So that kinda leaves Shanks out of that family. They maybe don't know who the mother is but only have a guess - and that guess is of the Figarland family(?).