The עדות ביהוסף, the מרכבת המשנה and the זיו משנה on the Rambam (Krias Shema 1:10) give the simple answer, that the Gemara (Brachos 9a) says that you can say Krias Shema until Neitz relying on the opinion of Rabbi Shimon Bar Yochai in time of need. But that only applies in a time of need, otherwise the halacha is like the Chachamim that it is simply not the time of Krias Shema. Note that they are explicitly arguing with the Kesef Mishna's understanding of the Chachamim's position.
It seems pretty clear that the Mishna Brura isn't understanding it their way.
The Kesef Mishna to the Rambam (ibid) says:
ונ"ל טעמן של דברים שהתורה אמרה בשכבך בשעה שדרך בני אדם שוכבין מסרה הדבר לחכמים שיפרשו עד אימתי הוי זמן שכיבה והם פירשו דעיקר זמן שכיבה הוי עד שיעלה עמוד השחר ולפיכך זמן קריאה בזמן הזה שוה לכל אדם אפילו למזידין ולפי שאחר שעלה עמוד השחר קודם שתנץ החמה איכא אינשי דגנו בההיא שעתא אמרו דאכתי הוי זמן שכיבה לגבי אנוסין אבל למזידין אמרו דכיון דרוב בני אדם קמו באותה שעה לא מיקרי תו זמן שכיבה
The reason for this is that the Torah says "when you lie down" - at the time that people lie down to sleep and [the Torah] gives it over to the sages to say until when the time of sleeping [ends], and [the sages] explain that the main time of sleeping is until dawn even for those who deliberately delay until then, but after dawn before sunrise there are people who fall asleep then so they said it is still a time of lying down for those who were forced, but for those who deliberately delayed they said since most people are waking up then it is no longer called the time of lying down.
The Divrei Yirmiyahu on the Rambam (ibid) says that this is a case where if you don't follow what the Chachamim say, even Min HaTorah you have not fulfilled your obligation. For evidence of this, he quotes Tosfos to Sukka 3a דאמר who says that one who eats in a small Sukka with the table outside the sukka in the house - which is forbidden as a decree lest he lean out of the sukka on to the table to eat - even Min HaTorah he has not sat in the Sukka.
The עמק הלכה from Rabbi Moshe Aharon Alufin, Number 5:6 says (after a long discussion justifying the view that Rabbeinu Yona holds that you are not allowed to say Shema after the time the Chachamim forbid it, like you are not allowed to blow Shofar on Shabbos Rosh Hashana - which seems to be the opinion of the Gra to SA OC 235) that the Chachamim set the time until Chatzos and you can't fulfill the obligation afterwards. The context there is according to the opinions that you can't say it after midnight, but the idea would seem to be equally applicable to not saying it after dawn.
For evidence of this he cites Tosfos to Bava Basra 48b תינח where he says that the Chachamim can uproot something from the Torah means they can make an action ineffective (i.e. Kiddushin) even when it was done explicitly without any stipulation of relying on the Chachamim, and against their wishes.
Any one of those last three views could be made to fit in the Mishna Brura (especially the first and the third).