I would be remiss if I did not mention the probably most famous explanation in this Rashi, offered by R' Naftoli Trop. He says there are two parts to Yisrael - shem yisrael and kedushas yisrael. Rashi holds this child has shem yisrael from birth, but lacks kedushas yisrael until geirus.
Rabi Akiva Eiger seems to pasken like Rashi, whom he references several times in Kiddushin and Yevamos; for example, see Gilyon Hashas Kiddushin 68b, and in his comments to Yoreh De'ah 266:12, that toRashi's opinion, we would not to bris milah of a goy sheba al bas yisrael on Shabbos, instead we'd push it off to Sunday. This implies he held to Rashi the kid is a full goy, not like Rav Ahron explained. What I don't understand is that in a Teshuva (#91) he says the opinion that the vlad is pagum (i.e., like Shulchan Aruch paskens), it may only be a problem dirabanan, but midioraissa she may marry a kohen. He makes no mention of his strict shita. After looking in Chiddushei R' Akiva Eiger to Yevamos 45 we see he held like Yam Shel Shlomo that goy haba al bas yisrael is only pagum midirabanan. So Rabi Akiva Eiger himself does not pasken like Rashi! He just feels that when Rashi says havlad kasher after geirus, means before geirus the child is a complete non-Jew.
Pischei Teshuva EH 4:1 has a decently sized discussion on this matter using completely different sources (except the Shaar Hamelech), and says this opinion is not accepted l'Halacha. Also see Otzar Haposkim 4:10.
Or Sameiach (IB 15:3) views the matter from a slightly different perspective. He says there are shitos which differentiate between thegoy or eved being boel a pnuyah, or an eishes ish Shulchan Aruch paskens they are the same). If the din applies to a pnuyah only, that the child is kosher, we see the mother's status is involved: this unmarried mother does not produce a mamzer (an unmarried woman could produce a mamzer through a case of incest, but that's not our case). But for a married woman the child would be a mamzer because the woman's status of married means this relationship was adulterous and the child is thus a mamzer. However, to the opinion (like which we pasken) that it doesn't matter if the mother is unmarried or married,the child is not a mamzer, we see the paternity is important. Non-Jews have no mamzeirus. Since this child is not a mamzer, the father's non-Jewish status must be passed down to the child in some way.