...a lot of theory went into the surprisingly effective American government. [...] So far as I know, there was no prior experience with large-scale democracy.

This depends crucially on what you're counting as large-scale democracy. The Roman Republic in some periods may qualify, although most of the time it seems to have been a de-facto oligarchy and its franchise was always quite limited. Iceland was governed by a representative body, the Althing, between 930 and 1262, but its population has never been very large. Venice had a (rather odd) electoral system during its city-state period. The development of the British Parliament from an advisory council into a full-blown representative body and major seat of government was extremely gradual and started quite early; Wikipedia cites De Montfort's Parliament in the late 13th century as the first elected one.

I think it's fair to cite the US under its current constitution as the first modern democratic republic of any great size, but I don't think I'd call it the first one.