After knowing that bad capacitors can destroy CWA ballasts and other ballasts with series capacitor circuits, I am beginning to wonder if bad power factor correction capacitors that are connected to HX autotransformer ballasts and simple series choke ballasts across the mains can destroy those ballasts.

If they are directly across the mains, they can not, at least not directly.

But it still can indirectly: E.g. when it bursts into flames, it will torch the ballast and the rest of the fixture.
Or if the lantern is used on some higher impedance source (like a generator or a long wiring), the eventual arcing may expose the rest to too high voltages and kill it that way. Or make the arc unstable so the fixture restarting very frequently, wearing off prematurely.
But these "indirect" ways are rather extreme and very infrequent, so in normal decent installation practically not happening.

Shorted PFC capacitor will certainly burn the primary of American style HX ballast, if not run directly at 277V, and if fuse/breaker would not react in a timely manner.

Fortunately, modern metallized film caps rarely fail with a short, they typically suffer from capacitance loss. It is actually quite hard to tell if disconnected/open PFC capacitor in HX autotransformer circuit can cause somewhat elevated current in some sections of the primary coil.


 

It is true that HX autotransformer ballasts are also common in other countries that use low domestic mains voltages such as Brazil, Taiwan, Japan, Mexico, Canada, Guatemala, Colombia, and a few others as well.

However, in many of those countries, you are more likely to find HX autotransformer ballasts in the form of preheat fluorescent tube ballasts rather than HID ballasts, but Canada and Japan have loads of HX autotransformer HID ballasts as well.

Fortunately, modern metallized film caps rarely fail with a short, they typically suffer from capacitance loss. It is actually quite hard to tell if disconnected/open PFC capacitor in HX autotransformer circuit can cause somewhat elevated current in some sections of the primary coil.

I tried a simple excersize:
Assume a 70W HX ballast with 240V OCV and two inputs, one 120V and second 240V, the PFC is connected to the 240V input tab, delivering 1A into about 75V load.
Assume common configuration, where the secondary starts from the 120V, so we get 3 windings, each with the same number of turns. I would guess not too far away from a 70W pulse-MH.
This way we have 3 winding sections in total:
  I - call it "primary", from N to the center tap (the 120V input), wound on the "primary" side of the core.
 II - call it "compensation" from the center to the PFC capacitor/"240V" tap
III - call it "secondary" from the center to the lamp output. This section is on the "secondary" side of the core, so behind the magnetic shunt, so exhibits high leakage inductance towards the rest.
Assume there is no gap in the main magnetic circuit, so the magnetizing current with no load is zero (a bit of simplification).

The secondary sees the full lamp current, so the 1A, in any case. Assume the lamp is good and on spec.

First the scenario with no PFC connected and feed from the 120V input (is first, because it is the simplest :-) ):
Because the secondary imposes 1A times the number of turns on the core and there is just the primary winding connected, the same 1A has to flow through the primary section too to cancel out the magnetizing current (no gap in the main magnetic circuit). So the mains current is then 2A.
So in this scenario we have
Primary with 1A
Compensation with none
Secondary with 1A
The ballast is not primarily designed to work this way.

Second scenario is fed from 240V input. Does not matter if PFC is there or not, as it is connected straight to mains.
Here the secondary field has to split among "primary" and "compensation" (here acting as a part of primary). Because the way how the windings are connected, the "primary" sees a difference between "secondary" and "compensation" winding currents.
If you put the equations together, you get
Primary with none,
Compensation with 1A,
Secondary with 1A
The ballast is designed to work that way.


The 3rd scenario is the most complex one, feeding 120V but the PFC capacitor present (assume set for PF=1 for simplicity).
Here the total 1A is handled by the "secondary", but then gets split to "primary" and "compensation".
Assuming the real total power input (so lamp power plus ballast losses) is 100W, so the mains current will be 0.83A.
The apparent power without compensation is OCV * 1A = 240VA. Half of this is handled by the "secondary", the other half needs to split among "compensation" and "primary".
So the reactive power for the PFC to handle is 218VAr, so 0.91A at the compensation winding.
The "primary" will then handle half of the real power, so about 0.416A.
So summary:
Primary has 0.415A
Compensation 0.91A
Secondary 1A
The ballast is designed to work this way.

So you can see the both ways the ballast is designed for, the secondary and compensation winding currents are about 1A, so both windings will be designed to handle that for most of the time. But the primary sees either no current, or just 0.41A.

But when fed from 120V with no PFC, the primary will see 1A instead of the 0.41A it is normally supposed to see, so technically about 2.5x current overload.

But what matters is not that much the currents, but the total power dissipation in the ballast, as that is the main factor determining the operating temperature. The power dissipation is I^2 * R for a given winding.
Now to minimize the total dissipation in a transformer, the amount of copper for each winding should follow the same ratios as is the VA each winding has to handle. And the winding resistance is inverse proportional to that cross section. Because we have all windings at 120V, ratios of currents will be the same as the VAs, assuming the manufacturer really optimizes to the last T and choses exactly those wire cross sections.
As the 120V input (with PFC as without is not the design target) is the worst case, we take:
So primary will have relative resistance 1/0.41 = 2.55,
Secondary 1/1 = 1
Compensation 1/0.91 = 1.1
So normal relative power dissipation will be 0.42^2 * 1/0.41 + 1^2 * 1/1 + 0.91^2 * 1/0.91 = 2.33
Same for the 240V input: 0^2 * 1/0.41 + 1^2 * 1/1 + 1^2 * 1/0.91 = 2.1
Now when removing the PFC: 1^2 * 1/0.41 + 1^2 * 1/1 = 3.44

So the power dissipation (so the temperature rise) will be about 1.47x higher than with the PFC. Comparing to plain higher current in all windings, it is equivalent to about 1.25x higher currents.

I assume American HX ballast will have much lower OCV, as small American HPS run at ~55V, and are designed to start readily from 120V American line voltage of 120V.

Next, the circuit is a bit more complex, with the secondary winding connected to its own tap at the autotransformer primary, below 120V line tap.

Also, phase angles for currents in the line, upper part of primary/capacitor, middle part of primary, lower part of primary, and secondary/lamp will be all different, so currents shall be calculated as vector sums.

For 55V lamps the ballasts use to be just series reactors, not HX transformers. That is in fact the main point of making the otherwise far from optimal 55V lamps in the first place - when budget is limited, the simple ballast has so much lower losses it offsets the lower efficiency of the 55V arc lamp.
But 55V waz is possible just for HPS, not for MH. MH are still in the 70..100V arc range, so need OCV in the 200..250V ballpark range.

Yes, the tap position in a real design is to be optimized, but the way depends on many factors, the operating performance (mass, efficiency, accuracy) being by far not the only factors.
The example I gave was intentionally made as a simpler configuration, not that well optimized for e.g. the shunt cross section (may need thicker shunt, so larger core or less space for the winding yielding thinner wire so higher losses, to share the same components or assembly jigs among multiple products,...) so total ballast mass and maybe cost, but still able to yield a working ballast.
But the real, optimized ballast designs are not that much off. What may be different is, some may make the "primary" intentionally thicker wire, so it can better handle the capacitor failure (as the failed capacitor may stay undetected, so keep stressing the ballast for a pretty long time).


And yes, you indeed need vector summing, that is what was has been done (otherwise 0.95A + 0.41A could not be able to compensate 1A in the secondary to yield the nearly zero magnetizing current of the ungapped main magnetic circuit)
What has been neglected are the effects of the higher harmonics of the arc voltage, but the rough idea of what is happening remains still valid...