When/How may I replace the Turn Coordinator with an Artificial Horizon?
The How It Flies in section on Spiral Dive Recovery mentions:

If you don’t have good outside references, you should not rely on the attitude indicator (artificial horizon). The attitude indicator contains a gyro mounted on ordinary mortal gimbals, which can only accommodate a limited range of pitch and bank angles. A steep spiral can easily cause the gyro to tumble, whereupon it will need several minutes of relatively straight and level flying before it can re-erect itself. Military aircraft have non-tumbling attitude indicators, but you’re not likely to find such things in a rented Skyhawk. Therefore, you should roll the wings level by reference to the rate-of-turn gyro.8 Being a rate gyro (as opposed to a free gyro) it has no gimbals, and therefore can’t possibly suffer from gimbal lock.
8 That is, the turn needle or turn coordinator, whichever you happen to have.

And in section on Spin Recovery it adds:

If you get into a spin in instrument conditions, you should rely primarily on the airspeed indicator and the rate-of-turn gyro. The inclinometer ball cannot be trusted; it is likely to be centrifuged away from the center of the airplane — giving an indication that depends on where the instrument is installed, telling you nothing about the direction of spin. The artificial horizon (attitude indicator) cannot be trusted since it may have tumbled due to gimbal lock. It is better to trust the rate-of-turn, which cannot possibly suffer from gimbal lock, since it has no gimbals. Remember, it is a rate gyro (not a free gyro), so it doesn’t need gimbals.

If you have good enough artificial horizon designed to avoid gimbal-lock, the you don't need turn coordinator. But with ordinary artificial horizon the turn coordinator combined with vertical speed indicator (despite it's problems like rather slow reaction time) are useful backup for recovering from in-flight upsets.