In every case of correctly operating CW transmitters, the RF output is interrupted to produce the pauses between elements, letters, and words.
To a modern ear, the term "Continuous Wave" is slightly misleading -- it doesn't mean the transmitter operates continuously, but rather it's in opposition to the "damped wave" that was produced by spark gap transmitters -- each spark started a pulse of RF energy, which decayed rapidly as losses in the RC circuit sapped the energy. With Continuous Wave, the oscillation has a constant amplitude while it operates; hold down the key, and you'll send a "continuous wave" until you release the key (or burn out something in the transmitter by exceeding its duty cycle limit).
The other misunderstanding here is that there's some other way to send than turning the transmitter off and on. If you stop the RF oscillator in a CW transmitter, you have stopped the transmitter from transmitting, just as you would if you interrupt the signal path from oscillator to amplifier, or cut the power to the amplifier stage. These will all reduce the RF output to zero, and it's switching the RF on and off that makes the "dots and dashes" of CW transmission.