The integrated error for a given unmeasured disturbance is inversely proportional to the controller gain. If you double the controller gain, you halve the integrated error. However, you need to keep the controller gain below a maximum that prevents oscillations from adverse changes in the process dynamics. The maximum controller gain corresponds to Lambda equal to the loop dead time (lambda factor equal to the dead time to time constant ratio).
The dead time from valve stick-slip or valve dead band is also inversely proportional to the controller gain so if you double the controller gain, you can halve the valve dead time, which further reduces the integrated error.
The improvement is only observable when there is a disturbance. Also, if the disturbance is very slow, small, or infrequent the integrated error from upsets may be negligible. For these cases, an increase in the controller gain has little effect on the standard deviation.
However, when there are a lot of upsets or set point changes, the increase in controller gain is important. A much smaller than possible controller gain is similar to additional dead time. For example, a lambda factor of one is equivalent to a dead time that is half of the process time constant. Stated another way, if the controller is tuned too slow, reducing the loop dead time by a faster valve, process, or measurement will not show a reduction in error. Nearly all studies on improvements in process dynamics retune the controller for a faster response to show the benefits.