MTAB
Your statement is inaccurate. You wrote: :because you could create mamzerim that way, the Torah says the husband has to give the get willingly, otherwise the get isn’t valid and they still are married, ”
That is not true. A get may be given against the will of the woman. It was a תקנת רבינו גרשם to disallow this practice. That does not invalidate the get, which על פי דין תורה is valid. Perhaps he instituted a consequence of cheirem in order to put teeth into this תקנה, since invalidating the get is not possible. Regardless, these is no issue of mamzeirus if a get was given forcibly.