Well, a couple of reasons really.
1 ) At jet speeds, the fin is more effective and as such naturally tends to resist slipping into the turn.
2 ) the yaw damper system detects the slip and applies top rudder.
and yes, it is primarily designed to counter Dutch roll, the autoco-ordination bit is a bonus.
Other than on a Jet Provost (no yaw damper) I have only ever had to use the rudders on jets to control the aeroplane on the take-off/landing roll or when asymmetric.
Trust me, it works!