minhag - Saying Misod and Reshut (introduction) for piyutim, then skipping them

I have seen that many yeshivos (and recently, even some shuls) often say just Misod and the Reshus on Rosh Hashanah and Yom Kippur, and then skip the actual piyutim they are meant to introduce!
This is even more evident in Musaf where they say Misod and go directly to Zachreinu L'Chaim. But it's clear that Misod is there because we are introducing piyutim in the first brachos (as is the case with other yotzros on yom tov and daled parshios). And therefore, on the second day of RH in Musaf, there is no Misod, even though we still say Unesaneh Tokef.

Question 1 - Does anyone know of any rationale for such a practice? (I understand that the goal is to save time in order to say other parts of the davening more slowly, but saying the first three piyutim (the ones that follow Misod) does not really add more than 2-3 minutes, (and they add so much to the davening.)

Question 2 - If they really need to skip parts of the davening, why not say less stanzas in Melech Elyon, and not lose out on the beautiful piyutim that connect the brachos that we just had a whole long Reshus ("Yereisi biftzosi" on first day) to introduce them?

Question 3 - And if they specifically don't want to say these piyutim, why say Misod and the Reshus at all?