You certainly couldn't call it a linear regulator, because the thyratrons are either fully on or fully off at all times. From the input's point of view it will exhibit negative resistance, with a reduction in the current required as the input voltage rises. That makes the classification especially hard to refute.
It's true that energy is being stored in the smoothing choke, but that alone doesn't make the difference since the choke could be used to accomplish the same thing in a traditional linear supply. The choke isn't inside the feedback loop, though, and that is a point in favor of the "It's not a switcher" camp. But it's the only one. In any event, charge-pump supplies don't need an inductor at all, and nobody questions whether they operate in linear or switched-mode.
(Edit: looking at the schematic, the filter choke is indeed inside the feedback loop. Case closed, it's a switching supply.)
That's exactly what happens in this power supply. The tubes charge up a giant (grapefruit-sized, if grapefruit were cubes) inductor, which supplies the current when the thyratrons aren't.
I don't see a good reason not to consider this a switching power supply. Phase angle is just PWM at a lower frequency.