Ho'il v'Ishteri only applies in cases where the Torah gives an exceptional "Heter" for something that normally should be prohibited - such as the Heter of Yibum Eshes Ach, or the Heter for a Metzora to put his thumbs into the Azarah for Haza'ah etc. It does not apply to the birth of a chick, since there is a very logical reason why birth removes the Isur of Sheretz and allows it to be Shechted.

The Beis ha'Bechirah (p. 44) offers us some direction. He explains that both the Isur of Sheretz and the Isur of Muktzah stem essentially from the fact that the chick is not yet born. Thus, the change in status from an unborn Sheretz to a born chick with regard to Shechitah launches a change to the status of a born chick for Hilchos Yom Tov as well - despite the normal laws of Migo d'Iskatza'i. Thus, if anything, the Sevara of Shmuel in our Gemara is similar to the Sevara in Sukah 7a, "Migo d'Havi Dofen l'Inyan Sukah Havi Dofen l'Inyan Shabbos." That is, if it has acquired the status of an acceptable Dofen for Sukah, it will acquire the status of acceptable Dofen for Shabbos as well - even though the requirements for Shabbos are normally different. (See Rashi's terse words here.)

I would suggest another Sevara. According to Rashi in Pesachim 47b DH v'Heichinu, the Isur of Hachanah is related to the fact that when a person eats an item that was not prepared for use on Shabbos or Yom Tov, he prepares it for Shabbos or Yom Tov by eating it. Preparing an object on Shabbos for eating is a Melachah that is prohibited by the Torah. (I refer you to my online Shiur [lecture] on Beitzah 2b for more on this subject.)

If so, Shmuel's Sevara is clear. Since the birth "prepares" a chick for Shechitah, the birth itself does the Hachanah of preparing it for Yom Tov as well. When a person later eats it on Yom Tov, he is not doing Hachanah since the birth already accomplished the Hachanah.

Best wishes,

Mordecai Kornfeld