rebdoniel: Chachmei Ashkenaz have spoken VERY highly about our piyutim, particularly those of Elazar Hakalir (who wrote the piyutim in bircos shma and almost all of the piyutim of chazaras hashatz on 1st day RH and YK). Tosfos and Rashi quote yotzros and other piyutim throughout (rashi on last week’s parsha actually just quoted one of the piyutim you wish to skip (see rashi on breishis 30:22).
I respect Ibn Ezra a lot (in fact, he wrote some beautiful piyutim himself including tzama nafshi that is to be said as an introduction to Nishmas but is now sung on friday night). But Ibn Ezra is NOT an Ashkenazi posek, and as such should not be used as a source for the minhag in an Ahkenazi shul.
The piyutim are extremely beautiful and are part of a SYSTEM. It makes absolutely no sense to skip the main ones that Elazar Hakalir wrote to say within the brachos and just scream out the “easy” ones after mechaye hameisim. Instead, one should follow the Mishna Brura (who by the way speaks very highly of piyutim and goes into detail also about the piyutim of Shalosh Regalim including the Maarovos said on Yom Tov night) and study the piyutim before Yom Tov.
As far as Misod, it makes ABSOLUTELY NO SENSE to say Misod and then skip to Zachreinu (although many yeshivos seem to do that). Misod is an introduction to the piyutim that immediately follow, if no piyutim follow then the Misod becomes a real hefsek since it serves no purpose but for the chazan to show off his voice.
And the best proof is that Misod is NOT said in Musaf of 2nd day RH even though Nesaneh Tokef, Vechol Maamimin and other piyutim are said later on. It is not said because no piyutim follow immediately. This seems pretty obvious to me, but please let me know if you have any better understanding of why Misod is not said on the 2nd day Musaf.
My suggestion: Get yourslef a machzor with a good perush. I suggest getting the new Koren Sacks. It offers excellent interpretations of the piyutim and lays out their beautiful structure in a way, that it my opinion, makes it a crime to skip them.
When you begin to understand the piyutim, and connect with hundreds of years of Ashkenazi mesorah (and the piyutim are remarkably similar on RH and YK between different Ashkenazi communities), I think you will get a different level of respect for Mesores Ashkenaz and will hopefully think differently.