If you send away the bird, why is Shiluach HaKen called that, meaning "sending the nest", and not "Shiluach Hatzipor", "sending the bird"? The nest is staying in place!
I think that the most straightforward answer to this question would be to suggest that, in Rabbinic Hebrew, קן ("nest") is also used metonymically to refer to the bird itself. So, for example, throughout Tractate Kinnim of the Mishna, where kinnim (קנים, "nests") is a way of referring to the birds.
There are numerous examples of rabbinic metonyms. Another would be "house" for "wife". In Shabbat 33b, to give but one example, Rabbi Shimon bar Yochai's wife is referred to as his "house". Jastrow lists this meaning as the fifth definition of בית. He also lists "birds" as a definition for קן together with its primary definition, the nest that they are in.
In R' Naftali Weinberger's book of Sefer "Shalai'ach Tishalach, A Practical Guide to the Mitzvah of Shiluach Kakan" (Published by Feldheim 2006) He writes in connection to the strange use and phraseology and says, "In response to this question, some explain that the term שלוח הקן is in fact a contraction of the phrase שילוח האם שמתיר את חקן sending away the mother bird, which permits the removal of [the contents of] the nest. For the sake of brevity and simplicity the Mishnah employs the first and last word of this phrase, resulting in the familiar appellation shiluach hakan." (no source is cited)