I'm not sure if it was out of inspiration from France or something else but presidential elections in Weimar Germany worked the same way - indeed, Paul von Hindenburg was not a candidate in the first round of 1925 (and of course was elected without receiving an absolute majority in the second round).
I don't think that Weimar Germany was inspired by the Third Republic... The Wilhemine Empire already elected the Reichstag in the same way - single member districts with two rounds, and probably open second rounds... And the same system is still used in many Swiss Cantons for local (and Ständerat / "Senate") elections - although this are often multi seat races elected at large, there is sometimes only one seat left open after the first round, and occasionally entirely new candidates might run in the second... (So, the second round is in fact FPTP - but only happens, if not enough candidates cleared the threshold for direct election - and often, parties that are closer to each other try to avoid competition by some horse-trading / withdrawing candidates...)
