The gemara[3] states that it is forbidden for a mourner during shiva to study Torah, because Torah study brings true happiness.[4] On Shabbat, only some elements of aveilut pertain; specifically, devarim sheb’tzina (practices of mourning that are not noticeable to others) apply.[5] Rishonim posit that Torah study is a davar sheb’tzina and is thus forbidden even on Shabbat.[6]
The Rosh and Shulchan Aruch[7] do indeed maintain that it is permitted for an avel on Shabbat to review the parasha, i.e., shnayim mikra v’echad targum. You suggest that this is because any set, regular learning is permitted on Shabbat; if only somewhat unique study is forbidden, perhaps daf yomi is permitted. However, the Rosh and the Shulchan Aruch explain their rationale differently, noting that since shnayim mikra v’echad targum is obligatory,[8] it is permitted for an avel to do it, just as an avel may and should recite Kri’at Shema daily even though it is a form of Torah study.[9] In other words, shnayim mikra v’echad targum is not viewed as an ordinary experience of learning Torah, which is improper for an avel, but rather is viewed as simply reciting things one normally recites in the course of the day.[10]
Some poskim, including the Korban Netanel,[11] do emphasize the matter of regularity regarding Torah study and extend the leniency to any standard, daily learning, even when it is done as actual learning per se rather than as a required recitation. Thus, he rules that those with the practice of reviewing eighteen perakim of mishnayot daily may do so on the Shabbat of aveilut. The Aruch HaShulchan[12] writes that based on this line of reasoning, those with a set quota of daily learning, including of gemara, may partake in it on the Shabbat of shiva. This should certainly include learning daf yomi.
However, this approach is not the mainstream one. For example, the Pitchei Teshuva[13] rules that those with the minhag to learn Massechet Shabbat every Shabbat should not do so during aveilut. The difference between that learning and the recitation of shnayim mikra v’echad targum is that there is no general expectation to take part in that specific Torah learning on Shabbat, whereas shnayim mikra v’echad targum is ideally supposed to be completed by Shabbat. (There is a parallel discussion regarding whether an avel should recite Bameh Madlikin on Shabbat.[14] On the one hand, it is a standard part of the siddur, but on the other hand, it is Torah that is recited not due to an actual obligation, but rather due to a weaker minhag.)
There may sometimes be other grounds to allow taking part in daf yomi on this Shabbat. There is precedent to allow certain types of learning because failure to participate in the learning would constitute conspicuous aveilut. The Shulchan Aruch[15] implies that while an avel should listen to the Torah reading, he should not receive an aliya even on Shabbat.[16] However, if he was called to the Torah, he must accept the aliya, because failure to do so on this Shabbat would be considered improper public aveilut. The Shulchan Aruch continues with the story of Rabbeinu Tam, who ruled that since he received shlishi every Shabbat, he should also get it during aveilut for this reason. Similarly, it is possible to argue that if one regularly takes part in a public shiur in daf yomi, or, for that matter, on any topic, he should not refrain from attending on the Shabbat of aveilut, as this can effectively constitute public aveilut.
However, it is difficult to consider refraining from participating in a regular shiur as public aveilut. The Taz[17] writes that although standard practice is to give an aliya to a father who is making a brit on Shabbat, nevertheless, on the Shabbat of the father’s shiva, he should not get one. It is rare that someone’s attendance at a shiur is more regular than is a father’s receiving an aliya before his son’s brit. If one always gave the shiur, his absence might be more conspicuous. Even then, however, one can argue that considering that the avel’s ability to prepare was curtailed during the week, his absence would not have to be interpreted as aveilut, but rather as a well-needed vacation, while allowing for his week-long stand-in to continue teaching. (Badei HaShulchan[18] suggests, in the context of whether or not to give the avel an aliya, that one should appraise each specific occurrence on its own.) In short, it is almost always improper to learn daf yomi on the Shabbat of shiva.[19]