One of the arguments that is heard when piyyutim are discussed, is that they are omitted totally by some, or at least not recited during chazoras hashatz or during other sensitive parts of davening, such as ברכות קריאת שמע, because they constitute a hefsek.
In a comprehensive shiur given in London a few years ago, Rav Hamburger discussed this argument. I am not going to get too deep into the shiur now (perhaps more later), but in regard to this specific point, he questioned, if those who say this believe piyyutim are really a hefsek, how can those same individuals and congregations say piyyutim during chazoras hashatz on ימים נוראים? Actually, come to think of it, he asked it with regard to saying selichos during chazoras hashatz during a taanis tzibbur vs. during neilah on Yom Kippur for example, but basically the point is the same, as piyyutim and selichos are basically the same. To take such liberties at ‘the holiest time of the year’? Are there any congregations who don’t do so?
Are there any answers to this question?