I believe that there is more than one issue in play. Your description of power transformer failure due to inrush current is certainly a valid and important issue for exactly the reasons you describe.
The issue described for the Heathkit Williamson family is always transformer failure, and the detailed discussions specified a primary short. Again, I've never heard of the filter cans failing and the replacement part was always a transformer. But your theory has some very interesting implications, so let's analyze what you wrote and expand upon it. Maybe we can rule it in or out.
I still think a piece is missing because we don't have reports of capacitor or rectifier failure which would surely ensue from the situation described.
A second issue is the high B+ voltage as the amplifier starts up. When the amplifier turns on the directly heated 5R4GY will fairly quickly (about 2 seconds) start conducting current and cause the B+ to rise quickly. At this time there is very little load on the B+ supply and as a result the B+ voltage will go well beyond (> 580V) the normal steady state operating voltage (510V) and stay at that voltage until the other tubes in the amplifier start conducting current, at approximately 8 to 9 seconds, at which time the B+ will settle down to the normal operating voltage (510V). This short time at elevated B+ can stress the G capacitors rated at 450V and coupling capacitors C14 and C15. I believe that this is what the Service Bulletin is referring to in the first sentence. "In a few cases it may be desirable to add a surgistor in the primary circuit of the power transformer to limit the high voltage surge which may occur when the amplifier is first turned on."
Yeah, I saw that wording and found it sloppy use of the vernacular. But the key point in all of this is that it is
not the capacitors which are reportedly failing, it's the
power transformer. Such failures could only arise from excessive current, as any excess voltage between the rectifier and capacitor would not burn out the transformer.
Hold that thought, I'm not in disagreement. I'm just stating the common theory we're going to test. I think I can explain how excessive voltage would blow the transformer's primary winding, or the B+ secondary. But I am not yet fully convinced this is also happening because the failures always describe transformers, not rectifiers or filter capacitors which also needed replacement. It always was
transformer, transformer, transformer, transformer, transformer, transformer, transformer, transformer, transformer, eh?*
Theory 1:
Heathkit's terminology is inapt and incorrect and thus deceptive. Describing current surges as voltage surges was common then and remains common usage; most builders in those days were not electrical engineers or physicists, and thus did not understand the difference between voltage and current. Hardly a shock, no pun in tended, as I have worked with electrical engineering PhDs who didn't know the difference and routinely misused the terms. (Digital and software backgrounds, unable to solder as well. Just saying that even with an engineering background the difference can be missed.)
So if the explanation is instead read with "current" substituted for "voltage" the above quoted material becomes:
"In a few cases it may be desirable to add a surgistor in the primary circuit of the power transformer to limit the high current surge which may occur when the amplifier is first turned on."
Which perfectly explains the inrush current from cold heaters plus the B+ to charge the filter capacitors. Simple and direct.
Plausability: High
Theory 2:
Excessive B+ blows the filter capacitors, which short, blowing up the rectifier and B+ winding. Suppose the tube cathodes using B+ take just enough time to heat that the rectifier is already conducting for about six seconds, the voltage climbs beyond the maximum voltage for the filter capacitors, the capacitors then short, the rectifier shorts (maybe it can survive, but damaged from over current), and the B+ winding burns out trying to drive a dead short. The already overhead primary may first fail, as it was primed for failure by the inrush current from the cold heaters.
The surgistor delay is about ten seconds so that theoretically keeps the B+ down until the tube heaters are hot. The heaters should be fully hot at about ten seconds, so that provides enough time for the overhead primary to cool and enough time for the B+ load (tubes conducting) to start so the capacitors don't short out.
This theory requires excessive B+ current to blow the winding. That requires high electrolytic leakage without a hard failure. I don't think a temporary leakage situation could be cured when the voltage dropped. If the anodizing layer failed, it would have a puncture through it that would not be self-healing. It's not like PIO where the short burns out and the non-conductive oil fills in the hole and the capacitor again functions.
Plausability: Low. This problem of initial overvoltage from rectifier on but matching B+ load not yet present exists in nearly every amplifier, but we don't generally see the filter capacitor bank shorting out and taking the rectifier and secondary to Valhalla. Yes, we do see this every once in a while when old filter capacitors fry. Great argument for shotgun capacitor upgrades to the power supply and a dim bulb tester for initial power on. The electrolytics are generally robust enough to tolerate a 15% voltage overload without failure. If they weren't nearly all of the tube amplifiers would regularly blow up the filters. That often doesn't happen even when output tubes are pulled and the B+ climbs to excessive levels. We don't have reports of capacitor and rectifier tube chernobyling in the Williamsons, either. What we do have reported are failures in the transformer. If a capacitor failure occurred it would have been reported as such, and it isn't.
I agree that the failure in a capacitor would blow the rectifier and winding; this is why I suggest separately fusing the B+
and the rectifier output because the mains fuse is sized for heater inrush and B+ charging, the total being high enough that a shorted rectifier or filter capacitor could take out the rectifier and then the winding.
My conclusion is Heathkit's wording is sloppy and it's a
current surge, not a
voltage surge.
Thoughts?
* The younger or foreign readers may not recognize this as the Olympia Café skit from Saturday Night Live in the early 1980s when it was funny.
for about twenty minutes and then 
You are braver than me!
