No, you can perfectly accelerate a Na+ ion in a weak electric field without exciting any transtions. Mass spectrometers and residual gas analyzers do that all over the place every day.
I also thinki it is a mistake in the book. 1/Gamma is not a time scale, it is an energy scale.
As for the mass-spring example, just try it out. Btw, it also works with a pendulum. If you wait much longer than the damping time, the everything will be in the ground state, no matter where you started. But if you move slowly compared to the oscillation period, the state of the oscillator will not change significantly, even if it is already in motion.