Okay, well, while seeking asylum in and of itself is not illegal, the fact remains that most illegals who are apprehended immediately claim asylum because it allows them to be released into the community while their hearing is pending and then they no-show. Claiming asylum is exactly and precisely following the script and training they were provided before coming and they follow it to a tee and en masse. So while claiming asylum is not in and of itself illegal- as you said- if a person apprehended entering the country illegally claims asylum- as they all do per the script- that does not negate the illegality of their entry.
It is illegal to enter this country unless you have received the permission of an immigration/border officer an an official port/place of entry. A visa overstay is not an illegal entry if you entered through customs/immigration. It does become a crime if your visa overstay generates an order to self-deport and you have not done so.
So - out my usual generosity of spirit- I will agree that an asylum seeker seeking entry through an official port/place of entry has committed no crime as long as they abide by the instructions of the immigration official or hearing officer. But again, claiming asylum does not make an illegal entry suddenly legal and out of the thousands who will be apprehended coming illegally across the border, the number who will not claim asylum will be mighty, mighty few.