"And how do you get from a sine wave produced by an oscillator to the sawtooth-ish thing needed to drive vertical sweep?"
I stand ready to be corrected, but I believe all sweep oscillators were pulsed. The RCA horizontal synchroguide was a blocking oscillator with a tuned circuit added to produce a sine-wave component for better reaction to sync pulses.
Oscillators in general do not produce pure sine waves naturally since their amplitude increases until something in the circuit limits it, so at the limiting element, the waveform is not a pure sine wave. An exception: HP linear oscillators used a light bulb as a slow-acting feedback element to set the amplitude below the point of non-linearity.
http://www.hpl.hp.com/hpjournal/pdfs...Fs/1960-04.pdf