Rogatchover-ish chakira found in the Likutei Sichos of the Lubavitcher Rebbe: Is the reason we don't say she'ehechiyanu on the last days of Pesach because --
1) Since these days are not considered a new Y"T there is no new obligation of she'hechiyanu.
2) Every second of Y"T, right through the last days, creates a new obligation to say she'hechiyanu, but that obligation is fulfilled through the she'hechiyanu recited on the first day.
(Compare with the machlokes in Chagigah 9 whether each day of Yom Tov is a new chiyuv of aliya la'regel or the chiyuv is fixed on the first day and each of the following days is just an opportunity for tashlumin.)
Here is the language of the Rambam (Shabbos 29:23) --
ובשביעי של פסח, אין מברכין שהחיינו, מפני שאינו רגל בפני עצמו, וכבר בירך על הזמן בתחילת הפסח
Notice that the Rambam does not just say the last days are not a new Y"T, but the Rambam adds that the bracha of she'hechiyanu was already recited at the start of the chag, implying that it is that original she'hechiyanu on the first days which is what exempts the final days from a new bracha.
Would a katan who becomes bar mitzvah in the middle of Pesach recite a new bracha of she'hechiyanu on the last days?