I was taught we fulfill the daily commandment to remember the exodus from Egypt at the end of the Shema, when we read:
אֲנִ֞י יְהוָ֣ה אֱלֹֽהֵיכֶ֗ם אֲשֶׁ֨ר הוֹצֵ֤אתִי אֶתְכֶם֙ מֵאֶ֣רֶץ מִצְרַ֔יִם לִהְי֥וֹת לָכֶ֖ם לֵאלֹהִ֑ים אֲנִ֖י יְהוָ֥ה אֱלֹהֵיכֶֽם׃
“I the LORD am your God, who brought you out of the land of Egypt to be your God: I, the LORD your God.”
— Bamidbar 15:41
Why is that the passage that fulfills the mitzvah and not any of the earlier passages in daily tefillah that recall Yetziat Mitzrayim?
The most obvious choice to me in Shacharit would be the Song of the Sea, which, practically speaking, is where I recall the exodus from Egypt most vividly each morning, given that it’s an extensive and descriptive textual reenactment of the very moment of leaving Egypt. Another choice could be the first parashah of tefillin, which is certainly as explicit a Torah reference to remembering Yetziat Mitzrayim as the pasuk in Shema.
וַיֹּ֨אמֶר מֹשֶׁ֜ה אֶל־הָעָ֗ם זָכ֞וֹר אֶת־הַיּ֤וֹם הַזֶּה֙ אֲשֶׁ֨ר יְצָאתֶ֤ם מִמִּצְרַ֙יִם֙ מִבֵּ֣ית עֲבָדִ֔ים כִּ֚י בְּחֹ֣זֶק יָ֔ד הוֹצִ֧יא יְהֹוָ֛ה אֶתְכֶ֖ם מִזֶּ֑ה וְלֹ֥א יֵאָכֵ֖ל חָמֵֽץ׃
“And Moses said to the people, ‘Remember this day, on which you went free from Egypt, the house of bondage, how the LORD freed you from it with a mighty hand.’”
— Shemot 13:3
לְמַ֣עַן תִּזְכֹּר֔ אֶת־י֤וֹם צֵֽאתְךָ֙ מֵאֶ֣רֶץ מִצְרַ֔יִם כֹּ֖ל יְמֵ֥י חַיֶּֽיךָ׃
“so that you may remember the day of your departure from the land of Egypt as long as you live.”
— Devarim 16:3
Rashi: זכור את היום הזה REMEMBER THIS DAY — This teaches that one must make mention of the Exodus from Egypt every day (Mekhilta d'Rabbi Yishmael 13:3)
The first page of this source sheet from Rabbi Yonason Sacks succinctly summarizes the relevant commentaries for the rest of my question. The Rambam doesn’t include zechirat yetziat Mitzrayim in his 613 mitzvot in part because for him it is incorporated into the mitzvah of kriyat Shema. There’s still wide agreement that zechirat yetziat Mitzrayim is its own rabbinic mitzvah, though, and the Ra’ah holds that while the evening zechirat is rabbinic, the morning one is Biblical.