“DY, he means to say that it beautifies the tefila like cufflings on a shirt or icing on a cake which should not be a hefsek.”
Reb Leizer, well put, that is exactly what I meant.
Furthermore, there are piyutim in the davening that are so integral to a specific tefillah that few even realize that they are piyutim. Ha’me’ir la’aretz is a piyut that contains the acrostic aleph-bet. The paragraphs preceding the birchot kriat shma (l’kel baruch, ahavah raba, etc…) are all piyutim. The fact that they are very early piyutim as opposed to later piyutim written after chatimat hatalmud doesn’t change their character. Mimkomcha in kedusha of Shabbat is a piyut, as are all the intermediate phrases of the kedusha of both shacharit and musaf on Shabbat (mimkomo, hu elokeinu, adir adireinu, etc…) . No one would dare say these piyutim are a hefsek.
Furthermore, the fear of hefsek seems to have been totally forgotten on tefilot yamim no’ra’im. The examples are so many and so well known they don’t require itemizing.
I find the hefsek argument to have too many indefensible holes, without even mentioning that piyutim are steeped in both the Ashkenazic and Sephardic mesorot. Kol ha’meshaneh yado al hatachtona.