Why shouldn't I take progesterone supplements throughout the entire pregnancy?

No risk or benefit. There is no reason that you can't continue it your whole pregnancy. It is of no benefit beyond 36 weeks but it won't do you or your baby any harm.
Progesterone. You can take natural Progesterone tablets during your entire pregnancy, but it's usually not necessary. The placenta makes >1000mg per day of progesterone, starting in the second trimester, so extra Progesterone isn't needed. You should not take synthetic Progesterone at all during your pregnancy (ex: provera) because it can cause birth defects and blood clots.