Shabbos Mincha: Sim Shalom or Shalom Rov?

Nusach Ashkenaz siddurim customarily have shalom rov as the selection for birkas shalom at Mincha on Shabbos, following the same custom as on a weekday.
The recently published Sefer Kush’yos, by a student of the Maharam mi’Rutenberg, asks the question: Why do we say sim shalom at Mincha on Shabbos and on fast days? Because the Torah is read on those occasions and one needs to mention, toras chaim, the Torah of life, which is part of the text of sim shalom.

This indicates that the early minhag Ashkenaz was sim shalom.

The Gemara says the reason why on ordinary days the priests do not ‘lift up their hands’ at Minhah is because of the likelihood of intoxication, but on the days [cited above] the question of intoxication does not arise.
(Ta’a’nis 26b. See also Orach Chaim 127, Hagahos Maimoniyos, Rambam, Ahavah, Seder Tefillos, Frankel edition p. 327))

It appears that there are two reasons behind saying sim shalom: 1) Torah reading which is referred to in sim shalom as Toras Chaim and 2) acceptance of the validity of having birkas kohanim at Mincha when there is no chance of intoxication by the kohanim i.e. fast days.