It's much more logical in France: no rabbit, the bells of Rome fly chocolates back from Italy to France, as one might expect at Easter.
(Some Americans protest that this is absurd though, since, after all, bells don't lay eggs [sic].)
P.S. It occurred to me, belatedly, that the whole "Easter rabbit" story is probably just a corruption of "Easter rabbi," a traditional name for Jesus.
Some evidence, mixing the textual original with the modified iconography:
https://www.walmart.com/ip/Ceramic-Mug-with-Easter-Rabbi-Hear-Eggs-Tulips/19891950110
Also https://en.uncyclopedia.co/wiki/Easter_Rabbit, but this seems to include some inaccuracies.
Similarly "cloches de Rome" might be a later allusion to the college of cardinals from a Gallican perspective; compare https://fr.wikipedia.org/wiki/Dumb_and_Dumber
