A PWM signal is a PWM signal is a PWM signal. So generating the right duty cycle from an MCU is trivial. You can add as many servos as you have PWM pins and even multiplex it of sorts with counters. See Servo.cpp in the Arduino HAL (there, I said it) for an example. Note that if you plan on doing sequenced animatronics, going with an MCU is the only sane solution. If you have a beefy MCU you can use the available pins or rely on a multi-channel driver like the PCA9685 previously mentioned. That leaves you overhead for more advanced processing like inverse kinematics.

There's a nice benefit of Servos I forgot to mention. Because they have a power pin, you don't have to worry about the output current of your driver. Any old logic high will work and the the servo manages it's own current draw.