Servo Driver
This is very possible, but it ain't easy.
Because your pulse width is so short (1-2msec) you don't have much time to be reading the incoming serial info. At 9600 baud it would take 1ms to transmit 10 bytes of data (start/stop bytes and 8 bytes of servo data). So every time you transmit to your servos, you may get a blip or hickup in your servos (not good).
Now if you setup the hardware UART with interrupts - you will have no problems. You can do plenty in between UART interrupts at 9600. I have controlled 6 motors with direction lines and series of LEDs that were pulse for ilum control (7PWMs and 6 +/- control) without problems.
Operating at 20MHz you should be good to send the appropriate servo PWMs on the PORTB for instance. Just setup 8 software timers and decrement them in time. As each timer goes to zero, load the new timer value (from the uart interrupt info), and flip the appropriate bit on portb. I would not recommend changing each port indiudally - setup a byte and change that byte - in the end write that byte to port b.
Attached is CC5X C code that I wrote to do this. It's big, but it may help.