Sort of right, when the horizontal oscillator starts, the horizontal output will go negative, and HV should start, if it does not, a dangerous state can happen and a free run overload with the output and it may start to red-plate and self destruct. Thus the need for the fuse M1 for protection.
The dependency on the vertical sweep to start after horizontal start depends on how the set is made, and the above mentioned boost voltage needed for vertical, which as mentioned this set does not seem to have, but it does seem to have a sync link via C120 R118 C4 from the plate of the horizontal output to the cathode of the vertical output.
So it looks like they are independent of each other, but if there was a problem in that above area, the vertical part linked to the horizontal might cause a problem?
But it more likely what EM mentioned, horizontal may be badly off, or even the waveform, meaning a problem with L38 B3 B1.or even B4.. YOU SUNK MY BATTLESHIP!!! :P