One final idea on the topic of she’hechiyanu on bedikas chameitz (part 1 here, part 2 here), and this one is admittedly a bit flimsy: perhaps the bracha of she’hechiyanu applies only to a chovas hagavra which applies infrequently, such as lulav, shofar, etc. The chiyuv of bedikas chameitz is not a chovas hagavra, but is rather a chiyuv on the house to be baduk. Since there is no chiyuv on the person, there is no bracha of she’hechiyanu.
A talmid chacham I told this to was willing to accept that bedika is a chiyuv on the home rather than the person, but remained ambivalent about whether this would impact the bracha of she’hechiyanu. My son is a contrarian and automatically rejects my Brisker chilukim, but he changed his mind a bit when he saw that R’ Baruch Ber also said that the chiyuv bedika is a din in the cheftza of the house. I haven’t seen the shiur (it is in one of the new compilations of R’ B.B.’s shiurim), but my son reports that the argument goes something like this: in order for person A to fulfill a mitzvah on behalf of person B, person A must be a bar chiyuva. For this reason a cheiresh, shoteh, or katan cannot recite birkas hamazon or blow shofar for an adult. R’ Baruch Ber quotes in the name of R’ Chaim that Reuvain also cannot put up a mezuzah on Shimon’s door for him. Since mezuzah is a chovas hadar, an obligation which rests on the resident of a home, and Reuvain is not a resident in Shimon’s house, Reuvain is just like the cheiresh, shoteh, or katan viz. a viz. the Shimon’s obligation to affix a mezuzah. However, Reuvain can do a bedikas chameitz for Shimon. Why the difference? After all, Shimon doesn’t have to search out an destroy all chamietz, only chameitz which he owns – the obligation is limited to Shimon’s property. If Reuvain is precluded from affixing a mezuzah for Shimon because residency in Shimon's home is a precondition to being mechuyav, Reuvain should be precluded from doing bedikah for Shimon because owning the chameitz in question is a necessary precondition to being required to do bedikah!
The distinction between the two cases is that mezuzah is a chovas hagavra, an obligation that rests personally on the resident of the home, while bedikas chameitz is not an obligation on the individual – rather, it is an obligation that rests on the home to be baduk. The requirement that a mitzvah being done by a bar chiyuva applies only when the focus of the mitzvah is the performance of the act; where the focus is the end result, the effect produced on the object, it matters not by whom or how that effect is achieved.
I thought my idea would answer a different question as well. Why is there no she'hechiyanu on the mitzvah of seforas ha'omer? Anon1 has been arguing in the comments that she'hechiyanu is only recited on mitzvos that bring about simcha (Tos Sukkah 46, echoed by the Ba'al haMaor with respect to sefirah done zecher l'mikdash), but the Rambam never mentions such a rule. Perhaps the answer with respect to sefirah is that again, there is no chovas hagavra to count; the chiyuv is on the days to be counted. This sevara would help us deal with the case of a katan who matures and becomes bar mitzvah in the middle of sefirah. Even though part of the count was done while he was a minor, since the focus of the mitzvah is days being counted, not who does the counting, the point is irrelevant. The (perhaps fatal) flaw in my reasoning is that based on my approach there seems to be no reason that a minor could not be motzi a gadol in sefirah. At least give me points for trying : )