The only primary source we found on the matter is the Tzitz Eliezer,[1] who infers from classical sources that kri’at haTorah and even U’Va L’Tzion must be at the proper time for Mincha. He is cited by Tefilla K’Hilchata[2] and Ishei Yisrael[3] without any dissenting opinion. (Ishei Yisrael also relates to a less conclusive oral ruling from Rav Chayim Kanievsky, who agrees that this is preferable.). The Tzitz Eliezer assumes that one should wait for Ashrei as well, although he does not cite support or even logic for that claim.
Let us analyze the matter based on the classical sources. Kri’at haTorah was instituted at Mincha of Shabbat due to the“yoshvei keranot.”[4] Rashi explains that these are businessmen who do not hear kri’at haTorah on Monday and Thursday, and so this is their additional reading. The Shita Mekubetzet[5] explains that the kri’at haTorah was instituted because many people get drunk during the day. The Rabbis placed kri’at haTorah along with Mincha to encourage people to come for Mincha, which is a time of divine goodwill and an appropriate time to show that we are different from those who get drunk. For some reason, the Tzitz Eliezer assumes that the reading itself has to be at a time when one can daven Mincha and may not precede it and lead into Mincha at the right time. He also cites the mishna,[6] which, after listing the various Torah readings, says that each one should be “at its time.”
It is not clear to all poskim that this short Shabbat kri’at haTorah has to be precisely at the same time that Mincha is recited. The Eshel Avraham[7] raises a doubt whether, if the need arises, one can do kri’at haTorah at a different time from when the group is davening Mincha. The Yaskil Avdi[8] writes that a group may do kri’at haTorah after people privately davened Mincha, and even after sunset.
It is a valid question whether the relating of this reading to Mincha tells us to (at least preferably) attach the reading to the tefilla of Mincha or whether it was instituted at the time of Mincha. If the former is correct, it makes sense to say that if it is done a few minutes before the time of Mincha, it should be fine. After all, P’sukei D’Zimra is meant to lead into Shacharit, and it can be recited earlier than Shemoneh Esrei can.[9] In contrast, if kri’at haTorah was instituted to be at the time of Mincha, it should ostensibly not be done any earlier than Mincha. It is also not clear if those who allow kri’at haTorah detached from Mincha maintain that the time is not so important. It is possible that they hold that the important thing is that kri’at haTorah be after the time of Mincha has begun, whether or not Mincha is being said at that time.
Your question assumes that the time noted on our calendars, half an hour after chatzot (astronomical midday), is the absolute earliest time for Mincha, but this assumption warrants investigation. The gemara[10] says that while Mincha is modeled after the afternoon sacrifice, which was brought half an hour after chatzot, Avraham Avinu would daven right after midday. Tosafot[11] and the Magen Avraham[12] are among those who say that conceptually, the time of Mincha begins at chatzot. We wait a half hour longer simply because we are concerned that we may recite it too early. It is not obvious to say that the Rabbis went so far as to push off the time of kri’at haTorah or Ashrei/U’Va L’Tzion due to this concern. There is also some question[13] regarding how to calculate the half hour (30 minutes or one twenty-fourth of daylight, which can be a few minutes longer or shorter depending on the time of the year). Thus, during certain times of the year, it may be possible, according to some opinions, to daven a little earlier than the time recorded on most calendars.[14] The case for leniency becomes even stronger when we consider that the Mishna Berura[15] cites opinions that after the fact, one who davened Mincha during the half hour after chatzot fulfills the obligation.
The Tzitz Eliezer relies strongly on kabbalistic sources that state that the spiritually appropriate time for U’Va L’Tzion and kri’at haTorah is the afternoon (Mincha time). Aside from the question of whether we are bound by such sources, since the time in question is after chatzot (and thus afternoon), it is very arguable that this time is appropriate.
We lack the conviction to rule against the stringent ruling cited without clear sources for leniency. However, we feel that since there are few sources and no compelling logic for stringency, one should certainly not oppose a minyan’s practice to start Mincha a few minutes “early.”