We repeat the amida in shul for the sake of those who do not know the amida. We read the birkat hamzaon aloud for the sake of those who do not know the blessings. In both cases, we assume that many people do not have a siddur or cannot read, and that they have not memorized the words. So, why do we not repeat the Shema in shul (and the blessings before and after), for the same reasons?
The Gra writes (Shenot Eliyahu to Brachot 1:1) that you have to read Shema yourself and you cannot fulfill your obligation by listening to another. (This is not universally agreed to, but see the next point.)
As for the blessings, the whole point of having a Chazzan starting from (just before) Barchu is for him to recite the blessings of Keriat Shema out loud to fulfill everyone's obligation. Like by Birkat HaMazon (and unlike Shemoneh Esrei), even someone who knows how to recite the blessings can listen and fulfill his obligation from the Chazzan. This is the agreed upon Halacha (Rambam Brachot 9, OC 59).
The Rosh writes in a responsum (4:19) that since the blessings of Keriat Shema are long and one might space out for a moment and miss some words, he would recited along with the Chazzan in an undertone (and finish a moment early in order to say Amen). (This is again just like many people's practice for Zimmun at Birkat HaMazon if they fear for spacing out.) The Shulchan Arukh quotes this recommendation approvingly (OC 59:4).
Nowadays unfortunately in many congregations the Chazzan is not careful to do his job. I know of no old source approving this change in traditional practice. (I have heard some argue that by reciting the openings and closings of the blessings aloud, Chazzanim have provided a minimally acceptable way for one to fulfill his obligation. This strikes me as a post-facto justification for the practice.)
BTW the reason we recite Birkat HaMazon out loud in a Zimmun is not because of people who don't know how to say it. It's to do a Mitzva publicly together (Brachot 45a). If someone there doesn't know how to say it, you can recite it for him even without a Zimmun (ShA OC 193:1).