[old_tv_nut]

One may ask, why not do away with the oscillator and just trigger the generation of a sawtooth on each sync pulse.
Two reasons:

1) An oscillator with a limited frequency range provides a great amount of noise rejection compared to a pulse-by-pulse trigger

2) In the horizontal sweep / high voltage circuits, the oscillator does not provide the sawtooth waveform. It is generated by switching a constant voltage across an inductor. If the current is not shut off in a reasonable time, it continues to increase until limited by the flyback primary winding resistance, a sure burn-out scenario. You may recall that the first IBM PC monitors could be destroyed if powered up without sync connected - the result of design by noobs who did not think of this problem, which had been solved in TV sets for decades.