The idea just never has come up before.
I believe that Midde'Oraita' the prohibition ceases when the animal dies, but Midderabbanan even when the animal dies the meat remains forbidden forever. Someone who knowingly ate meat from an animal that was still alive when dismembered, regardless of how long afterwards, would be in violation of this commandment.
Furthermore, Halakhically for Noachide purposes death occurs when all limbs stop moving (other than just nervous spasms). Thus unfortunately there is no way of knowing whether most meat in the non-Jewish world really is permitted or not. But as I understand it, one is obligated to abstain only when one is sure of this and it is not the burden of the eater to know this for certain.
For Jews an animal is permitted immediately after shechitah, and most authorities hold that kosher meat is permitted for non-Jews.
Fish don't even require killing to be eaten. They are permitted immediately upon removal from the water.