Joseph:
You wrote: A Get requires neither a rabbi nor a Beis Din. All a Get requires is a Sofer and two eidim.
That is correct. Now, this woman receives her get. Who will verify that everything was done in accordance with halacha? She will probably want to remarry. Would you accept her for a daughter-in-law? Or would you want to know that someone with the halachic expertise verifies the get as kosher? That is why it is common practice to use a beis din (either a sitting BD, Zabl”a, or the get BD I described). The kashrus of the get affects the husband as well, as he has the Cherem DeRabbeinu Gershom issue over his head. Unless he got the heter meah rabbonim, the get needs to be known to be kosher.