“The following is a proof that in the רשות הרבים the women who wore wigs completely covered them with a kerchief:”
You keep on going around in circles; this is taken out of the opposing shita i.e. the words of the Bee’r Sheva. The Magen Avroghom and all others saw his taynes and sided with the Rema and SH”G – Halacha has been decided – not like the Be’er Sheva – period!