Fortunately I had the honour on multiple occasions to do public kiddush on Friday nights in different synagogues. A few of them were keen on asking me to recite the three faceted blessing after it, although I've rarely seen anyone else doing this. Food was not served after the public kiddush. A similar issue is the grace after meal leader's cup of wine. Which sources discuss this obligation?
Shulchan Aruch Orach Chayyim 269 rules that if one recites kiddush in a synagogue without a meal following it (where such is the custom) the adult making kiddush should not drink the wine but instead give it to a child to drink.
Mishnah Berurah 269:1 writes that if no child is available, the adult reciting kiddush should make sure to drink a revi'it so that it counts as having made kiddush in the context of a 'meal', and to afterwards recite the three-faceted blessing.
As far as the cup of wine drunk by the leader after grace after a meal, Shulchan Aruch Orach Chayyim 190:2 rules that he should indeed recite the three-faceted blessing. (In 190:3 he writes that the leader ought to drink a full revi'it of wine in order to ensure that he is definitively obligated to recite the after-blessing.)