Many have discussed that in oldprayerbooks Psalm 30 is not printed before Barukh Sheamar, and some have claimed that its addition in Nusach Ashkenaz to that point is probably a mistake, a possible misunderstanding of a Kabbalistic practice, especially in light of Tur Orach Chayim 281:
ויותר נכון מנהג אשכנז לומר כל המזמורים בין ברוך שאמר לישתבח כדי שיהא לכולם ברכה לפניהם וברכה לאחריהם.
And the Ashkenazi custom of saying all the psalms between Barukh Sheamar and Yishtabach is more appropriate, because for all of them there would be a blessing before and after.
It is also related that the Vilna Gaon was against this addition (Maaseh Rav 26). However, since the 19th century in almost all Nusach Ashkenaz prayer books we find Psalm 30 there. Which authorities argued that it should be still said and how did they reason?
The Aruch HaShulchan (cited below) writes that Psalm 30 before Baruch Sheamar is part of Nusach Ashkenaz. He also offers an original reason for its recitation. He writes, "the practice of reciting the verses of Mizmor Shir Chanukas is based on the Gemara (Shabbos 30) that says that when Shlomo placed the Ark into the Holy of Holies he was only answered when he mentioned the merit of David. Since the verses of Pesukei D'Zimrah are the verses of King David, Psalm 30 was added beforehand and not in the middle because in it David's righteousness is portrayed.
כ' בערוך השולחן (או"ח נ:ג-ד) וכו' בנוסח אשכנז אחר ברייתא דר' ישמעאל אומרים מזמור שיר חנוכת ואח"כ ברוך שאמר ואח"כ הודו וכו' וטעם אמירת פסוקים אלו דמזמור שיר חנוכת אמרו שלמה בשעה שהכניס הארון לבית קודש הקדשים ולא נענה עד שהזכיר זכות דוד כדאיתא בשבת [ל'.] ולפי שכל פסוקי דזמרה הם פסוקי דוד המלך לפיכך הקדימו מקודם מזמור זה שבו נגלה צדקתו ולכן לא הכניסוהו בתוך פסוקי דזמרה [נ"ל]
After having posted my question I've done some research and found some additional points:
R' AvrahamLandau in Tzelota deAvraham discussed that Psalm 30 was not present neither in Siddur Rav Amram Gaon nor in older Ashkenazi prayer books, and it contradicted the words of the Tur in Orach Chayim 51. He then explained the Kabbalistic background of the practice, which was introduced later to the Ashkenazi practice. But he still recommended its recital, because thus we express our wish that the Temple be soon rebuilt in our days, and if we intended but couldn't perform a commandment due to circumstances beyond our control, it is considered as we have actually done it (Berakhot 6a). The same is true for King David, who wanted to build the Temple, but he was not allowed to do so (Chronicles I 22:7–8). Moreover, by saying a mourner's kaddish there, we also express that Psalm 30 belongs to the recital of the daily sacrifices and not the Pesukei deZimra.
Tzelota deAvraham cites the Rema in Orach Chayim 25:13 saying that it is a Kabbalistic tradition to say 4 kaddish with a tefillin on, and by saying a Kaddish deRabbanan after Rabbi Yishmael's baraita, a mourner's kaddish after Psalm 30, a half kaddish before Barekhu and a full kaddish after the repetition of the amidah, even those fulfil this requirement who are in hurry and have to leave after the repetition.
As mentioned in the other answer, the Arukh haShulchan mentions that the gates of the Temple didn't move until King Solomon mentioned the merits of King David (Shabbat 30a). By inserting a kaddish one can also separate Psalm 30 recited by King Solomon (see Bamidbar Rabbah 12) and Pesukei deZimra attributed to King David.
I've found a site citing the Likutei Mahariach, the work of R' Yisrael Chayim Friedman of Rahó, who also repeated that Psalm 30 was not present in old prayer books, and one reason to say it before Barukh Sheamar is that this later addition should be separated from the text compiled by the redactors of the siddur.
See also the entry in Otzar haTefilah on pp. 21–22 by R' David Kahn.
See also this article in Yeshurun by R' Yitzchak Satz regarding the different types of kaddish.