Overmintmarks show that either an incorrect punch was used and corrected, or that dies made with the intention of sending to one mint were changed and sent to another. Such as S mintmarks nickel reverse dies still on hand at Philadelphia at the end in 1937 or 1954 were repunched with the D when it was decided that San Francisco would not strike nickels the following year. If the branch mints could add mintmarks there would not be correspondance asking for either permission to use unmintmarked dies that were received, or permission to have them replaced with marked dies. Specifically I am thinking of the wires from San Francisco to Philadelphia about the unmintmarked dies received in 1870. Follow up correspondence from Philadelphia told them not to use them and that replacements would be sent. San Francisco replied that they had already cut an S into the gold dollar and three dollar dies. And if you look at the 1870-S three dollar gold you will see that the S is NOT punched into the die, but engraved.
We shall have to agree to disagree for there are too many accounts of one branch mint sending dies already having their mint mark to another branch mint that had run out of dies. And when those dies arrived, that branch mint punching their own mint mark over top the other existing mint mark. Yes dies are hardened steel, but punches are hardened steel too. And hardened steel can punch into other hardened steel.