Are the passages following the blessings over the Torah always mandatory before prayer?

Question

The Ashkenazi Artscroll Siddur, from which I learned most of my liturgical Judaism, has a page in the very early sections of שַחֲרִית for blessings over the Torah. These are blessings which it claims must be recited before reciting any portion of the written or oral Torah. Here are the two blessings (abbreviated):

... בּרוּךְ אַתׇּה יי ... וְצִוׇּנוּ לַעֲסוֹק בְּדִבְרֵי תוֹרׇה
Blessed art thou Hashem ... and who has commanded us to engulf ourselves in words of Torah

בּרוּךְ אַתׇּה יי ... אֲשֶׁר בׇּחַר בׇּנוּ מִכׇּל הׇעַמִּים וְנׇתַז לׇנוּ אֶת תּוׄרׇתוׄ
Blessed art thou Hashem ... who chose us from all the nations and gave us his Torah

Immediately following these two blessings are three passages, one from the Torah (Numbers 6:24-26), one from the Mishnah (Peah 1:1), and one from the Gemara (Shabbat 127a). In the explanatory notes at the bottom, it says that once a blessing over a mitzvah is recited, the mitzvah must be performed immediately. Therefore, the three passages, which include all forms of Torah written and oral, should be recited.

The question I have is what procedure should/can be followed when praying in shul. Clearly, the morning prayers contains passages from the Torah, so the blessing is required. However, is it required that I recite the three passages? Consider the following scenario. If I arrive at shul, recite the two blessings over the Torah, and immediately jump into either the offerings or פְסוּקֵי דְּזִמְרָא then would I not be reading excerpts from the Torah thereby fulfilling the blessing? In this case, would I not be required to read the three excerpts of Torah? Or, would it be the case that I should still recite them to make sure I have covered all my bases (Torah, Mishnah, and Gemara)?

Answer 1

According to one site, there is a machloket

The minhag is to say the Pesukim of Birkat Cohanim and the משניות of Elu Devarim after Birchot HaTorah. If one didn’t learn then, it’s a dispute whether one fulfilled one’s obligation and so one should have intent to fulfill one’s obligation with the Bracha of Ahava Rabba

↑ S”A 47:9-10, Mishna Brurah 47:19-20

Another site discusses the case where a man receives an aliyah before saying the passages:

If a man comes late to shul and receives an aliyah, but hasn’t said Birchat HaTorah yet, what should he do? The Torah obligation is to say Birchat HaTorah in the morning. The brachot before the Torah reading are only m’drabanan. The Kitzur rules that if one can recite the Birchat HaTorah of the morning quickly, one should do that prior to receiving the aliyah. If it isn’t possible, one can accept the aliyah and recite the Birchat HaTorah of the morning afterwards. However the blessing of Asher bachar banu should be omitted.