Some of the Rishonim (e.g. Shibolei haLeket) discuss whether there were other solutions R’ Chiya could have utilized to make up the bracha shortfall. What about answering amein to the brachos of kri’as hTorah, the haftarah, etc.? Would that count towards the total of 100? Aside from the issue specific to these brachos of whether there is an obligation on the listener to recite them, which is then fulfilled through shome’a k’oneh, or whether the bracha belongs to the person getting the aliya (see R’ Yosef Engel in Tziunim laTorah), the more general question is whether one can fulfill the obligation to recite 100 brachos by listening and answering amein. Some Achronim opine that since R’ Chiya chose to eat snacks rather than rely on this option proves that one must recite 100 brachos – shome’a k’oneh will not work. Others suggest that R’ Chiya was in a situation where there may not have been a kri’as haTorah and he was therefore forced to avail himself of snacks, but had he been able to do so, he would have relied on shome’a k’oneh.
The Tur (O.C. 46) writes that even though the shaliach tzibur recites the birchos ha’shachar in shul, one should independently recite them as well. The Tur then goes on to quote this din of reciting 100 brachos. The Ba”CH suggests that the Tur quotes the din of 100 brachos here as the justification for each individual reciting the birchos ha’shachar. Shome’a k’oneh, just hearing the brachos from the chazzan, is not enough to meet the quota of 100 – one must recite the brachos independently. The Beis Yosef, however, learns that when the Tur writes that the brachos must be recited by the individual he simply means to tell us that they are not a chovas ha’tzibur, but must be said by each individual even where no tzibur and chazzan are present.
What’s the lomdus behind whether shome’a k’oneh should or should not work here? You could say that the question hinges on the general issue of how shome’a k’oneh works: is it as if the listener has really said words, or is it an alternative mechanism to being yotzei even without saying words? The machlokes Rashi/Tosfos (Brachos 21) as to whether one can pause and listen to keduah in the middle of one’s own shomeh esrei is generally assumed to depend on this question. Does listening to kedushah constitute a hefesk because it is as if one has interrupted one’s own davening with speech, or is shome'a k'oneh a mechanism to be yotzei even without speech and therefore is no hefsek?
Perhaps we don’t need to deal with the broader question of how shome’a k’oneh works and can limit our scope to the particular takanah of saying 100 brachos. The gemara quotes the source for saying 100 brachos as a derasha/asmachta from the word “me’ah,” as we saw above, but there is a tradition quoted in the Geonim (quoted in Tur 46) that David haMelech instituted saying 100 brachos to avoid deaths from a plague. If we look at the takanah as just some formal rule, then shome’a k’oneh may work just as well as actually saying a bracha. But if the point is to ward off a plague, then it would seem likely that it is the closeness to Hashem that results from having made the effort to recite 100 brachos and not the formal recitation itself which is key. It is the personal effort to actually recite the brachos, not merely to hear others engage in saying brachos, which is crucial, and therefore shome’a k’oneh would not suffice.