Interesting mareh makom: The Meiri writes in Magen Avos (Inyan #23) that when Purim fell on Shabbos/Sunday the minhag where he lived in Provence was to fast on Friday, not on Thursday as we do. He presents the argument not to fast on Friday as being based on two factors: 1) even though a ta'anis chovah can be fasted on Friday, ta'anis Esther is not a "chovah gemurah;" 2) the principle of avoiding "akdumei pura'nusa" that would prevent the fast from being moved up a day does not apply to ta'anis Esther because that fast is really a day of simcha.
Meiri has a simple proof that even a minhag is a good enough mitzvah reason to allow a fast on erev Shabbos or erev Yom Tov: ta'anis bechoros.
The Mishna Berura in O.C.686 writes that we do not fast ta'anis Esther on Friday because the added time for davening would take away from time spent preparing for kavod Shabbos. I saw someone ask why the M"B cites this as the reason instead of quoting the halacha (O.C. 249, which the MG"A the M"B is quoting refers to!) that one is not allowed to take on a voluntary fast on erev Shabbos/Y"T. As Rama paskens, ta'anis Esther is not really a "chovah gemurah," so Fridays should be out. In light of the Meiri (which I don't know that the M"B saw), perhaps the M"B did not feel that this sevara enough.
The Divrei Chaim (the real one -- the Sanzer Rav) was so busy on ta'anis Esther he could not even speak with his talmidim (see here). Just as in the days of Esther it was this day which was the day of preparation for war against Amalek (see first Rosh in Megillah quoting R"T) through tzom and tefilah, so too in all generations, on this day in which we can ask for rachamei Shamayim to defeat Amalek. It's a day to be busy on!