Why just by adding water or more wine does it not make it not pagum any more?
Here's a kabbalistic view on why to add water to the wine (fun fact: in Dutch, we have a proverb called "water bij de wijn - putting water into wine" - taking compromises).
The Zohar says in parashas Shmini that wine represents gevurah and that (shining) water is on the side of chessed:
אֶלָּא שֵׁירוּתָא דְּחַמְרָא חֶדְוָותָא. סוֹפֵיהּ עֲצִיבוּ. וְעוֹד דְּיַּיִן מִסִּטְרָא דְּלֵיוָאֵי אָתֵי, מֵאֲתַר דְּחַמְרָא שָׁרֵי, דְּהָא אוֹרַיְיתָא וְחַמְרָא דְּאוֹרַיְיתָא, מִסִּטְרָא דִּגְבוּרָה הוּא. (ס"א סיפיה עציבו, וכהנא שירותא וסיומא בעי חדותא ונהירו דאנפין, יין אתי מסטרא דליואי) וְסִטְרָא דְּכַהֲנָא מַיִין צְלִילִין נְהִירִין.
for the Torah and her wine are of the side of Gvurah, and the side of the priests is that of clear shining water, WHICH IS CHESED. (translation taken from here)
I saw an quote from the Ta’amei Haminhagim (not sure where, if anyone could add this in...) in saying that when you add water to the wine that is pagum, it will become "ready to drink again". This is because when you add chessed to gevurah, it will give you berachos, blessings.
See also the Tosfos ד"ה טעמו פגמו, where the Yerushalmi is quoted. The Tosfos says:
And he puts some water into the cup, so that it will appear to be unblemished. As it is stated in Yerushalmee: R” Yonoh tasted the cup and he corrected it by adding water or wine and the explanation is that after he drank from it he added a drop of water or wine to correct it and then he would recite havdoloh or kidush upon it. (emphasis mine)
Tosfos says that this way, it will appear to be unblemished.