I don’t know if that makes it mutar, but it certainly makes it much less of a chilul shabbos.
grama is still assur and chillul Shabbos. If we are dealing with an issur d’rabbonon (which according to many, we are), what difference does it make that you call it “less” of a chillul Shabbos?
since a light or a fan is attached to the house, by completing the circuit you are doing boneh.
I don’t think it has anything to do with being connected to a house. Most understand the Chazon Ish to mean that you were “boneh” the circuit.
BTW, check the other thread for the source of the Ramba”m.