pasuk which stands as Rabban Gamliel's prooftext, baavur zeh, also establishes a chiyuv for a person, in each generation to view himself as if he himself went out of Egypt. Or alternatively, to conduct himself, showing others, as if he personally went out of Egypt.
It is a question of vocalization. לראות could have a chirik under the lamed, as pictured, in which case it means "to view himself", or it could have a patach under the lamed, in which case it is shorthand for the Rambam's alternate nusach of lehar'ot, to show himself.
Why personally? Because it states asa Hashem li, "to me", and betzeiti, "when I went out."
The pasuk, meanwhile, is cast as saying this in the future, when they have arrived and settled in the land of Israel. It was stated before the entirety of that generation was condemned to die in the wilderness, so perhaps one could argue that the was actually only targeting the generation that left Egypt, who could personally say this. Especially if we view it not as a chiyuv but as a promise to an enslaved people.
However, viewing it as an everlasting statute and chiyuv, it stands as a basis for talking about leaving Egypt in first person. How can a person do this? As stated earlier, had they not been redeemed, behold we, our children and our grandchildren would still be enslaved to Pharaoh.
"Hashem also brought us out of there." Isn't this just another prooftext saying the exact same thing as before? I don't think so. Rather, consider the context:
From Devarim 6:
It could have been someone who was actually present for the Exodus who spoke this to his son. So what does this teach us? Well, realize that there are two parties to this conversation, the father and the son. The father was present. We can grant that. But the son was not present, or else he would have had no cause to ask. Yet, the pasuk says וְאוֹתָנוּ הוֹצִיא מִשָּׁם, "and He brought us out of there," rather than veOti hotzi misham, "and he brought me out of there." Thus, the son who was not present at the Exodus is included in וְאוֹתָנוּ.
(Of course, there is ambiguity here. In some languages such as Hebrew and English, the first person plural can be inclusive. If inclusive, it includes the addressee (that is, the second person). If exclusive, then it is speaking about plural people, where the speaker is a member of that group, but the second person addressee is not a member of that group. In the derasha, as opposed to in the peshat, the word וְאוֹתָנוּ is taken as inclusive.)
Thus, going back to the words of the Haggadist: Not only our fathers (such as the speaker in Devarim 6) did Hashem redeem, but also אוֹתָנוּ, us, the sons, did He redeem together with them. This was a very deliberate phrasing by the Haggadist which reveals the structure of his derasha.
And so, the first prooftext was from the perspective of the father who was not actually physically present, and the second prooftext was from the perspective of the son.
One we have internalized, or externalized, that we indeed have been redeemed, the logical conclusion is that we have an obligation to praise Hashem. This is what bridges between Maggid and Hallel.
And perhaps this is an aspect of the chiyuv of sippur, to relate the transition from gnus to shvach, as it states: from servitude to freedom, from sadness to joy, from mourning to holiday, and from subjugation to redemption, and we say before Him, Hallelukah.
This closes Maggid and, for now, this commentary on the Haggadah.