The obligation to bring the wood offering was a consequence of the family’s vow.
When they made that pledge originally, it binded them for all future generations. When deciding the legal boundaries of a vow, we consider the mindset of the person at the time of the vow. In our scenario, when a family vowed to bring the wood offering, they knew it was common that the day could coincide with Shabbos, and took it as obvious that the previous family would want to use their entire time-frame for their offerings. Therefore, when the vow was originally made, it was made with the understanding that if the day fell on Shabbos, their obligation would be moved to the following day, Sunday.
Both of Rashi’s explanations, therefore, are compatible. When he says that, on Friday, the time for the offering had not yet arrived, he is referring to the fact that implicit in the vow was the stipulation that when the day fell on Shabbos, the family would move to Sunday to accommodate the previous family, as Rashi detailed in his commentary to the Rif.
This all applies to a situation where one family’s vow invades the time of another family’s. That is when we defer their obligation to Sunday. But when no such conflict exists, there is no reason to defer one’s commitment to later. So, for example, when a yahrzeit falls on Shabbos, there is no reason to defer a vow of charity to Sunday. Rather, one can advance their charitable giving in honor of the yahrzeit to Friday, for there is no reason to delay the fulfillment of a mitzvah.
