I don't know if they are technically called "piyutim", but there is a nusach we say on Rosh Hashana/Yom Kippur between "Ata Kadosh" to "Hamelech Hakadosh". Was that decreed by the Anshei Knesses Hagedola, or was it a more modern text? If it is more modern, how is it not a problem of a hefsek?
Rambam (Seder Tefillos Mikol Hashanah) calls it מנהג פשוט, a widespread custom, to say these paragraphs (his version is different than ours, but it's recognizable). So it doesn't sound like it goes back to the Anshei Knesses Hagedolah; perhaps it's from the Geonim. (It appears in Siddur R. Amram Gaon, but there are lots of later interpolations there, so I don't know that this proves anything.)
It wouldn't be a problem of hefsek, because you are allowed to insert extra prayers for communal needs even in the first three berachos (Shulchan Aruch, Orach Chaim 112:1). Certainly, these paragraphs are no worse than the other insertions in the first and last couple of berachos (זכרנו, מי כמוך, etc.), which are allowed.