The gemara (4b) tells us that bittul alone is sufficient to remove any issur of chamietz, but Rashi and Tosfos disagree as to why this should be true. Rashi explains that since the Torah uses the general term "tashbisu" instead of specifying any specific method of destroying chameitz, bittul suffices to fulfill the mitzvah. Tosfos disagrees and holds that tashbisu can only be fulfilled by destroying the chameitz or burning it according to Rabbi Yehudah. The reason bittul removes the issur chameitz is not because tashbisu has been fulfilled, but because bittul makes the chameitz hefker, and one is not responsible to destroy chameitz which is hefker.
The Minchas Chinuch suggests that Rashi and Tosfos disagree over the fundemental definition of the mitzvah of tashbisu. According to Rashi, the mitzvah is an issur aseh which is fulfilled so long as one avoids the possibility of owning chameitz. According to Tosfos, tashbisu is a real aseh that is fulfilled only through a kiyum b'yadayim of actually destroying chameitz.
Returning to R' Akiva Eiger's answer, according to Rashi, just like bal yera'eh is not violated by chameitz found during Pesach so long as one has made all ncessary efforts to remove chameitz beforehand, so too the issur aseh of tashbisu cannot be violated. Only according to Tosfos can one argue that if one discoveres chameitz, even though bal yera'eh has not been violated because one has already done bittul, there still exists a mitzvah b'yadayim to destroy the actual chameitz if one is unwilling to declare it hefker.
Perhaps another dimension to this machlokes revolves around the understanding of bittul as hefker. The Rambam in Hil Nedarim (2:14) famously writes that hefker works like a neder that one accepts not to benefit from the mufkar property. If bittul is just a form of hefker, R' Yosef Engel asks, why are we not concerned with the possibility of sha'aila which would revoke the hefker status - in fact, using the principle of ho'il, just the possibility of revoking the neder should render bittul ineffective (the Rambam writes that it is assur to revoke a neder of hekdesh, but many achronim hold that if one ignores the issur and does revoke the neder it can be undone)? According to the Rambam, one is forced to learn that bittul is far more than ordinary hefker - it is tanatamount to an actual destruction of chameitz, a fulfillment of tashbisu, just as Rashi explains.