Note: All sources were from the shiur Shalom Zachar- What's the point? Why on Shabbos? Why that name? on YUTorah.
While the reason quoted in the Taz you brought above (as well as several reasons given by other commentaries) does indeed apply to girls as well, there are also several reasons given for a Shalom Zachor that apply exclusively to boys (including a different reason given by the Taz a few lines earlier). It would seem that viewing the totality of all the reasons, the Rema concluded that the custom of a Shalom Zachor applies only to a boy and not to a girl.
Here are several reasons why the Shalom Zachor applies only to a boy:
Taz 265:12 (earlier in the piece you quoted):
וראיתי סמך אחד לזה ממדרש רבות פרשת אמור פרשה כ"ז ר' לוי אמר משל למלך
שגזר ואמר כל אכסנאין שיש כאן לא יראו פני עד שיראו פני המטרונא תחילה כך
אמר הקב"ה לא תביאו לפני קרבן עד שתעבור עליו שבת שאין ז' ימים בלא שבת
ואין מילה בלא שבת עכ"ל.
In summary, this meal is an 'extension' of a Seudah for a Bris. Since having experienced a Shabbos is a requirement for having a Bris, we celebrate on the Shabbos as an extension of the Bris.
ספר מגדול עוז - יעב"ץ
He paraphrases the Medrash that says the fetus is commanded in keeping the all the Mitzvos. Since Shabbos is the first Mitzvah that the baby 'encounters', a feast is made to remind him of his obligation (note that this would also apply to a woman, and while the יעב"ץ does ask about why there isn't a 'Shalom Nekeivah', he brushes off the question by saying it applies more to men since men are obligated in more Mitzvos).
תשובות והנהגות כרך ב סימן רב
ומיהו נראה שעיקר הטעם שמשום בשבת הוא ע"פ מה שמבואר בזוה"ק שכל השבוע
מתברך משבת קודש, ולכן יש חשיבות לשבת שלפני הנשואין או לפני מילה או
אפילו לפני יאהרצייט כי אז הזמן שהשפע כבר מוכן, רק יורד כל דבר בזמנו,
ולכן בשבת מודה אבי הבן ומשבח לה' על השפע וברכה דמילה שבא כבר אז בשבת,
וכל אחד בא ומברך אותו, ומסייע זכות הרבים לזכות בברכה ושפע, ומודים
ושמחים ומשבחים אלקינו ית"ש על חסדי
Similar to the Taz above, R' Moshe Shternbuch says that the Shabbos before the Bris has a special holiness, and we make a gathering for people to give blessings and receive the special holiness.
Admittedly, there are also several reasons (in addition to the one mentioned in the question) as to why we celebrate a Shalom Zachor that would seemingly apply to girls as well. However, in practice (at least among Ashkenazi Jews), it seems that in spite of the below reasons, we only make a Shalom Zachor for boys:
תרומת הדשן סימן רסט
ואע"ג דמפרשי אינהו בשם ר"ח דהיו רגילין לעשות סעודה כשנולד זכר על שנושע
ונמלט ממעי אמו
Essentially, this is a Seudas Hoda'ah for the child surviving the dangerous trauma of childbirth.
נפש הרב [הר' צבי שכטר שליט"א] עמ' רמ"ב
In summary, a woman within the first week of giving birth is considered someone who is deathly ill, and if there is any need to be Mechalel Shabbos for her wellbeing, we do so. The Rambam in Hilchos Shabbos 2:3 writes that when desecrating Shabbos for the sick, it should be done by 'scholars and sages of Israel' (seemingly in order to stress the importance of saving a life). Once we had the scholars of the generation going over to the home of the woman who gave birth, we made it into a big party.