Is it safe to drive a 5V servo using a logic level shift board?
Many problems. That circuit is a level shifter for data signal, so it is not a power switch. Basically if it tries to turn on a servo, the high supply return currents would go into MCU IO pin, so it would not work, and could even be damage the MCU.
Also even if the transistor is used in a proper way to disconnect servo ground, the problem is that when a device has both data and power supplies connected to it, it is not a good idea to disconnect ground, but the positive supply.