Based on my understanding...
One is not allowed to make preparations on the 1st day of Yom Tov for the 2nd day. An Eruv Tavshilin allows one to cook, bake and make food preparations on Friday for Shabbat (whether Shabbat is Yom Tov or not.)
Is someone allowed to roll a sefer Torah on Friday Yom Tov for using that Torah on Shabbat? Preparing the Torah in advance avoids tircha detzibbur. However, I understand that the eriv tavshilin only allows for food preparations, or would it also allow for something like prpearing the Torah as well? Or, is the avoidance of tircha detzibbur, for some reason a way of allowing this advanced preparation?
While, I don't know if this is a factor, here, but, say that there is no extra Torah available so that they can prepare one for Shabbat reading before Yom Tov begins. Would this situation make any difference in halacha?