It’s not about a Chashash Chimutz. It’s a Gemara in Temurah (maybe Me’ilah) that a calf that grew up its entire life eating only Issurei Hana’ah is Treif (the Rishonim discuss a Zeh V’zeh Gorem issue and what if it eats both Tarfus and other stuff; there is also a strange Taz which extends it to all Ma’achalos Asuros, not just Issurei Hana’ah). So the issue is that if the cows and chickens eat Chametz, the milk and eggs could have a status of something that grew only because of Issurei Hana’ah. (And when something is Assur B’mashehu we say that Zeh V’zeh Gorem is Assur, so there’s that issue too. If you assume that the cow’s nourishment and body combine to produce the milk, which we Halachically do, then the milk that was produced because of nourishment from Chametz on Pesach should be Assur.)
We cannot be M’vatel Chametz before it becomes Assur because we hold that since Chametz is an Issur Mashehu, it’s Chozer V’neor on Pesach and becomes Assur, even if it’s .00001% of the mixture when Pesach starts. Eggs and milk are an entirely separate issue, as I explained above. The reason not all eggs and milk are Assur on Pesach is because since the Chametz was only a Zeh V’zeh Gorem and is completely gone without a trace before Pesach, then we say that it was completely gone was Pesach started, not just Batel.