People very strongly emphasize eating the entirety of an eruv tavshilin on Shabbat (some even designate an honoree to eat the eruv). No one I have ever heard of, however, ensures that an eruv chatzerot is eaten. Why is that?
What about eruvei techumin?
The Eruv Tavshillin works by beginning to prepare for Shabbat before Yom Tov starts. If you don't plan on eating the food on Shabbat, then you didn't really start your preparations using the designated food.
Eruvei Chatzerot and Techumin use the food to mark the collective residence of the enclosed area or to mark one's "primary" dwelling place for the Shabbat respectively. Even if you don't plan on eating the designated foods that Shabbat, they can still accomplish those goals.
That said, the Rama notes (OC 394:2) that in places where the Eruv Chatzerot is eaten and replaced each week, it should be used for the Lechem Mishneh on Shabbat, a place of honor for the object which was used to perform a Mitzva.
Eruv tavshilin gives you permissions to cook for shabbos because you need to cook for shabbos and you are showing it by starting to cook and bake even before yom tov started. So those 2 foods are your proof that you started cooking before and that you need them for shabbos. Therefore, you should eat them on shabbos to finalize your proof.
Other eruvs are there as just in case your get hungry, you can eat them or not makes no difference. Imagine you need to take a long walk on shabbos and you are not sure you'll make it to the destination so you call someone who lives in the middle and you tell them, if im stuck and cant make it to my destination can i drop by your house to rest and to get a bite to eat? But if you dont actually stop by their house to use their services you still stayed calmer knowing that you have a place to make a stop over.
