Matzos because of Haste

At seudah shlishis today, a friend of mine asked why the reason given for Bnei Yisroel taking matzos with them is the lack of time necessary for the bread to rise brought about by the haste in departing, given the issur chametz. I tried to answer based on the Mishnah in the 9th perek of Pesachim (96), which states that Pesach Mitzrayim did not have the issur chametz kol shiv'a that Pesach l'Doros has, but forgot the conclusion of the gemara that aligns this mishnah with a drash of R' Yosei haGelili, that the issur still extended the entire first day.

After the fact, I wondered if there was any significance to the fact that the ba'al hashmu'ah of that (apparently accepted) shita is R' Yosei haGelili, who also holds by the unaccepted shita that there's no issur hana'ah by chametz at all (28b), so that in his mouth, the din would only be referring to an issur achila that spanned the first day, but it seems more likely that when the rabbanan accepted this drash, they interpreted it k'shitasam that even the issur hana'ah would have existed on the entire first day. At any rate, RYhG is coming to be meikil, so that even if we were to reduce the scope of his drasha, it would not achieve the desired result.

Although the drash of RYhG is based on the passuk "v'Lo yei'acheil chameitz", we hold like Chizkiyah (21b), who reads this passuk as referring to an issur hana'ah, so there doesn't seem to be a way out on that path, either.

To sum, then, it seems clear that there was an issur of owning chametz on the night of 15 Nissan until the end of the day. That being the case, why could we not say that the reason for their not carrying chametz with them was because of the issur, rather than using the practical reason of the haste that the passuk specifies?

R' Yitzchok Dovid Frankel asks a related question in Machat Shel Yad, and develops the idea that chipazon was an intrinsic part of Yetzias Mitzrayim, so that despite their ability to avoid the rush by anticipating the Yetziah earlier that day, there was a necessity that chipazon be associated with Yetzias Mitzrayim, based on HaShem's haste in taking us out at exactly the necessary time. This being the case, perhaps this could also serve as a reason for the Torah's pragmatic rationale for the matzos, rather than the latent halachic rationale.