When shivah ends on Shabbat, does it end when Shacharit ends on Shabbat or the time Shacharit would have ended if it were a weekday? Since we’re talking about a community in which people do not pay shivah calls on Shabbat, should we go by the time that services end on weekdays? If so, if we start start davening on Shabbat after we normally finish davening during the week, could someone whose last day of shivah was Shabbat receive an aliyah in shul?
As you note, Rema rules in Shulchan Arukh YD 395:1 that even if מנחמים don’t usually come on day 7 (which, as the poskim note, is our practice on Shabbat), one still has to wait until after the end of davening, since that is the time that the consolers would come on other days. There is a problem with the logic here, as the last day of שלשים ends immediately with the morning, since no consolers are coming, so why should it not be the same here? Achronim suggest we do so to keep things consistent, but it remains a difficult pesak, and there are poskim who disagree with it.
Your question is a good one. Since Rema frames this as “waiting until the time you would wait on the other days of the week for the consolers to come, after they leave shul,” it seems that we are to look at the rest of the week, which would mean the non-Shabbat davening. And, as we said, some poskim disagree with Rema altogether. So, yes, I think this mourner can consider his aveilut done once it has passed the time of morning davening during the week, and can get an aliyah.