Placed right after Ma Nishtana, it constitutes an answer to the question of the reason for these strange practices. This is what the gemara in Pesachim 116a states:
מתחיל בגנות ומסיים בשבח: מאי בגנות רב אמר מתחלה עובדי עבודה זרה היו אבותינו [ושמואל] אמר עבדים היינו אמר ליה רב נחמן לדרו עבדיה עבדא דמפיק ליה מריה לחירות ויהיב ליה כספא ודהבא מאי בעי למימר ליה אמר ליה בעי לאודויי ולשבוחי א"ל פטרתן מלומר מה נשתנה פתח ואמר עבדים היינו:
Thus, according to Shmuel, the way one begins with genus is by saying Avadim Hayinu. And when Rav Nachman (a student of Shmuel) and Daru his servant fulfilled the obligation of asking (or informing so as to ask), he immediately went on went on to Avadim Hayinu.
And it seems that the shevach is also there, in the same statement, that Hashem took us out of there with a strong hand and outstretched arm. The bit of us still being there if not for this redemption is a good way of appreciating the importance of this, and why we should be thankful, and perhaps also a bit of חייב אדם לראות את עצמו...
Just as we saw before (when discussing Mah Nishtanah) that there were many ways of fulfilling the asking of the question, and being exempt from Mah Nishtanah, so too there may be several ways of fulfilling the genus to shevach. And so this might be a machlokes in which everyone wins. Or in which we could theoretically fulfill this retelling using a number of different texts.
But while this short answer might be enough (except perhaps for an obligation to mention and explain Pesach Matzah Marror), this text tells us that we should go on at length about the exodus from Egypt. Even those who already know it all, there is this Mitzvah of going on at length.
But then, we don't go on at length about Yetzias Mitzrayim, at least not immediately! Rather, we go on at length about our obligation to say over, and to go on at length! Thus, the Haggadah continues:
Some say that the roles of these particular individuals, as descendants of converts, Leviim who were not subjugated, etc., we might think they were not obligated, or not so obligated. And so we see to what extend they did it.
I would simply say that these were a group of chachamim, navonim, yod'im et haTorah, etc. And so it directly illustrates the preceding, and the extent to which one should relate the story of the exodus, because Harei Zeh Meshubach. (That it goes until Krias Shema shel Shacharis, I recall a suggestion that this despite the fact that Zecher Yetzias Mitzrayim is part of Krias Shema, and so we see this is a separate obligation. This is cute, but this is just a mark of the time, that they spent the entire night discussing Yetzias Mitzrayim. There are other Biblical aspects of the obligation of Krias Shema shel Shacharis. But see what I write below about Lod.)
Now, obviously, all these Chachomim couldn't have said the text of the Haggadah as we have before us. After all, how could they relate a story about themselves? It hadn't happened yet! That night, it seems, they were mesaprim about the actual Exodus, not the halachos about their obligation to relate.
On the other hand, we have this incident in Lod, in the last Tosefta in Pesachim:
חייב אדם [לעסוק בהלכות הפסח] כל הלילה אפילו בינו לבין בנו אפילו בינו לבין עצמו אפילו בינו לבין תלמידו מעשה ברבן גמליאל וזקנים שהיו מסובין בבית ביתוס בן זונין בלוד והיו [עסוקין בהלכות הפסח] כל הלילה עד קרות הגבר הגביהו מלפניהם ונועדו והלכו [להן] לבית המדרש
Here, the obligation is to engage in the hilchos hapesach. In a nice parallel to the above -- just as those Chachamim were engaged in sippur yetzias Mitzrayim and then transitioned to zechiras Yetzias Mitzrayim, Rabban Gamliel and the Zekeinim were engaged in discussing hilchos haPesach all that night (and viewed this as the obligation), and transitioned to go to the Beis Hamidrash where they continued learning, perhaps other things. (Perhaps one can argue that these are the same...)
What is meant by "Pesach" in Hilchos HaPesach? This Pesach night occurred in Lod, post-Churban Habayis. Two possibilities strike me. The first is the halachos of the Korban Pesach, as we see in the same Tosefta Pesachim, but in perek 4, with what they asked Hillel Hazaken (as I would understand it). And learning something is like doing it, just as we learn/recite korbanos every morning. Perhaps we could then view this as reenactment of the Korban Pesach by learning it. Alternatively, the gemara discusses learning Hilchot Hapesach, HaAtzeret, and HaChag 30 days before, so perhaps these are general rules of the holiday.
This discussion about the obligation to discuss might be a response to the Chacham (or, in other variants, the Tipesh), to tell him ke-hilchot haPesach ... ain maftirin achar hapesach afikoman. Perhaps different parts of the Haggadah address different sons, and this is chanoch lanaar al pi darko.
One final thought -- despite staying up all night learning the halachos of Pesach (as in the Tosefta), or staying up all night discussing the exodus from Egypt (as in the Haggadah), they did not neglect eating the Matzah, Maror, or meal. They did not neglect drinking the four cups of wine! If students would interrupt them for a tangential mitzvah, like Shema of the morning, they would certainly interrupt them to let them know that they were not going to fill the mitzvos of the night. Unfortunately, people often neglect the other mitzvos of the night in favor of Maggid. Forget about the obligation of finishing by chatzos halayla. At least, this is a machlokes. But in some houses, they extend Maggid such that the children, or young adults, fall asleep and do not drink the four cups! (Meanwhile תניא רבי אליעזר אומר חוטפין מצות בלילי פסחים בשביל תינוקות שלא ישנו.) Divrei Torah for the child's Rebbe, or that you yourself want to say over is nice, but when you optimize one thing, it is often at the expense of another. Save this for Shulchan Orech, for the meal. Or save it for after Nirtzah; or for the Yom Tov meal the next day.