As I understand it, an Eruv Tavshilin works by making it as if the preparations for Shabbos began before Yom Tov started, and it works as one long preparation from erev Yom Tov through the entire first/second day of Yom Tov.
If this is the case, why can’t the eruv be used to allow preparations on the first day for a second day of Yom Tov, even when there’s no Shabbos involved?
Because you can’t cook on Shabbos, but you can cook on Yom Tov if the Yom Tov falls out on a weekday.
The Gemara (I believe in Be’ah) is of the opinion that theoretically, the 2nd day (biblically) is like *chol* in which case you would be insulting/desecrating the first day by preparing for the 2nd day.
This is why we wait late at night to start 2nd day (neros, etc…) so as not to come to desecrate the first day.
When the 2nd day is Shabbos, we have to finish prep. by a bit before sunset, because Kedushas Shabbos is highest compared to Kedushas Yom Tov.
If we are taking on that the first day is kodesh, any preparation for next day is preparing for “chol” which you may not do on yom tov, even with an eruv tavshilin.
How are we allowed to cook on Chol Hamoed for Aharon shel Pesach (or for Simchas Torah), which (from the Torah) are definitely considered yom chol?
Nothing to do with eruv tavshilin, right?
Chazal determine which melachos may be done on chol hamoed, and they said it’s muttar.