I am answering this question using the material in the link posted by the OP. The interpretations are mine.
Proxy marriage originated in Europe in the Middle Ages. It was meant for 1) "power couples," such as kings and queens, or very high-ranking nobles, 2) who wanted to ally, but lived in widely separated geographical locales, and 3) who did not have convenient means of transportation.
First, it does not seem to have been usual during Greek or Roman times (if it existed at all). In the days of those empires, the power was centralized in the "capital" e.g. Rome or Athens, meaning that "power couples" consisted of marriages between the "boy and girl" next door.
It was in the Europe of the Middle Ages, after the collapse of the Roman empire led to the creation of numerous and widespread power centers, that kings and queens felt the need to go outside the country to find a suitable match. Given the "patchwork" of Europe's feudal
holdings and states, the country next door might not be a good ally, but rather an enemy, and a spouse with common interests (or enemies) might be someone from two or three countries over. This would be a determination made by the diplomats, with the sovereigns assenting. Possibly one or both would be too preoccupied with war or other matters to visit the other, so they would make the marriage (and resulting alliance) by proxy.
In modern times, we have two-career "power couples," but air, land, and ship transportation is so much better that they don't have the "commuting" problems of the European nobles. If anything, proxy marriage in America is for the LOWER classes, particularly where the man is engaged in the military, or some other "wandering" profession, and has trouble connecting with his intended bride.