Perhaps we need to start by defining what the word eved means in Judaism anyway. It doesn't necessarily mean slavery or servility; in the Bible it is frequently used of royal ministers (and even in one instance - see the answer I linked - King Rehoboam is advised to "be an eved to the people"). Great figures in Jewish history - Moses, Joshua, David - are described as "the eved of G-d," and more generally, the Jewish people collectively are called His avadim.
In all of these cases, the idea is that the person so described is devoted to a higher purpose, and becomes elevated thereby. (The Talmud, Shevuos 47b, comments that "the eved of a king is himself like a king.")
In the era of Moshiach, "the occupation of the entire world will be solely to know G-d" (Rambam, Laws of Kings 12:5), and in particular, "the Jews will be great sages and know the hidden matters, grasping the knowledge of their Creator according to the full extent of human potential" (ibid.). So while "the Sages and the prophets did not yearn for the Messianic era in order to have dominion over the entire world, to rule over the gentiles, to be exalted by the nations..." (ibid. 12:4), the point is that the peoples of the world will themselves realize that it is an honor and a privilege to be "servants to the servants of G-d" (a title used by R. Akiva Eiger, one of the great sages of the past couple of centuries) - to do whatever they can to create the optimal conditions for "G-d's avadim" to do their work.