First of all, I don't really buy the premise that elections were often limited to city states mainly because of logistical problems. I would rather argue that it was because the polis was the primary societal identification for most freemen in the Mediterranean lands, and therefore it was natural for the people of each city-state to want to govern their own affairs. Recall that the Greeks were organizing Olympic Games between the various Greek city-states as early as the 8th century BC. It is hard to imagine there to be a far step from there to holding elections, if they had so wanted to.
As to when, more precisely, "large scale" elections first took place, that's a bit of a matter of definition. To begin with, clearly, in Ye Olden Tymes, countries were overall far lesser in terms of population than they are today, so in that way it isn't really comparable at all. It also depends a bit on just how "centralized" you want to specify the elections to be, and the nature of the assemblies being elected. Here are some examples of elections which affected larger areas:
- The classical example is probably the English parliament, whose Commons were being elected by the boroughs from, I think, the 14th century or so (though I admit I don't know more precisely how the local elections evolved).
- The Riksdag of Sweden was first convened in 1435, following elections in the local hundreds (which had a previous history of sending representatives to the ancient things of the various provinces).
- The old Swiss Confederacy was established in 1291 by representatives of the founding cantons. I cannot find any precise details on exactly how the Eidgenosse were bestowed with the authority to swear for the entirety of the cantons, but it is likely that they were chosen by local Landtage or similar.
- The Third Estate of France seems to have been elected with rather wide suffrage from the 15th century onwards, though I know very little about the details.
- The Holy Roman Empire had complex (non-centralized) systems of representation, some of which were elected. For instance, the Free Imperial Cities elected local burgers to represent the cities at the Imperial Diets. While mixed, of course, with hereditary nobles, it's not as if the Diets didn't have elected components. Likewise, while the election was between a very small number of Kurfürsten, it should be said that the Holy Roman Emperor himself was elected by geographically widely separated electors.
- The Christian Pope has clearly been elected from the very beginning by the bishops and/or cardinals (depending on era) of the entirety of Christendom, clearly spanning a very large geographical area. Likewise, the Ecumenical Councils were constituted of representatives from the whole of Christendom.
Some of these, particularly the English Parliament, the Swedish Riksdag and the French Estates, have evolved in rather straight lines to their modern-day descendants. Exactly when you want to draw the line for them being a "large election" is up to you. :)
There may be many more examples outside of Europe, too, but I'd be less knowledgeable about them. My spontaneous reaction to the answer you linked about China is, however, that it seems somewhat simplistic. I very much doubt China was quite as integrated as both that question and the answer make it seem, and I would not be the least bit surprised if there were assemblies of representatives of various levels organized in different ways at different times. I really wish I knew more about this.