Why doesn't a full yoke deflection also change the stabilizer trim on the Boeing 737 MAX? Various websites have stated that during the 737-MAX crash in Indonesia, the pilots could not pitch up because of a faulty AOA sensor pitching the plane down.

The stabilizer has a rear fulcrum and front jack screw. The elevator, positioned by yoke, is hinged to the stabilizer.
If the yoke is pulled back, the elevator moves up. Then the stabilizer must move down, either automatically or manually, to remove the yoke loading.
However, if the yoke is fully pulled back, why doesn’t this force the stabilizer down directly so that the pilot always has total control of pitch without checking the flight manual or pushing switches?
The yoke does move the elevators, whereas the trim moves the horizontal stabilizer. The stabilizer is much larger than the elevators, and therefore is more effective than the elevator at extreme angles. Even with full backpressure, the nose would continue to pitch down. 



Nonetheless, all 737NGs/MAXs have trim cutout switches, 1 for the autopilot and 1 for the normal electric trim. Using these would immediately disable the electric trim. These are generally part of a standard trim runaway situation, but the crew did not recognize the situation and failed to use them.