Kaddish before the blessings of Kriyat Shema when there were not ten

A congregation that began Shacharit without a minyan, and after Yishtabach, before Yotzer Or, the tenth person arrived, can they recite the Kaddish before the blessing of Yotzer Or?

The Taz in Orach Chaim, Siman 55, Se'if Katan 3, writes: "But certainly if it happened that the minyan did not arrive until after Ashrei, this should not prevent saying Kaddish, and this is the custom every day when reciting Tehillim in the morning without a minyan, that Kaddish is said when the minyan arrives, and this is correct." However, the Magen Avraham in Siman 234, Se'if Katan 1, writes: "If Ashrei was recited without a minyan and then a minyan arrived, they should recite one psalm and then say Kaddish. And it seems to me that the same applies when learning in the synagogue without a minyan and then calling ten to say Kaddish, this is not proper, and it is a kal vachomer, because the main reason for Kaddish is the sanctification of the name, and therefore ten are required during the learning, as we learn from 'and I shall be sanctified among the children of Israel'. See Siman 157."

The Mishnah Berurah in Siman 55, Se'if Katan 2, rules like the Pri Megadim that Kaddish after Pesukei Dezimra is more obligatory than Mincha, and therefore the Magen Avraham agrees with the Taz that Kaddish can be said even if there were ten only for Kaddish. However, the Mishnah Berurah in Siman 53, Se'if Katan 11, writes: "And in a case of necessity, if they also completed the blessing of Yishtabach and then a minyan arrived, they should say at least three verses from Pesukei Dezimra, and the shaliach tzibur can say Kaddish, because otherwise Kaddish cannot be said since there were not ten during the recitation of some verses of Pesukei Dezimra or Yishtabach at least." From his words, it seems that if they were during the recitation of Yishtabach, it suffices. And so it seems from Se'if Katan 8. In Siman 54, Se'if Katan 8, he writes that Kaddish always refers to the verses and not to Yishtabach. And Kaddish should only be said on the verses.