It's not the first time this has been brought up. It's a LM317AHVT regulator which is a 1.2V - 57V part and is being used beyond spec. With the low current output though it helps its cause.

BTW, the value of the R1 and R3 resistors on the schematic are incorrect. From the following thread showing pics of those resistors, R1 is 576 ohms and R3 is an 82.5k ohm resistor.

https://pinside.com/pinball/forum/topic/help-with-sten-galaxy-please#post-4014225

This sets the output voltage somewhere around 182.5 volts.
The maximum input to output differential spec for this regulator is 60 volts. If the input to that regulator goes over 240 volts, it's in trouble.
To reduce the HV output from the transformer, make sure it's not wired for 110VAC.

yes... until that one burns up, too.

A non-"AHV" part would probably be more prone to burning up than the AHVT part.
The LM317AHVT has an *absolute* maximum input to output differential of 60 volts.
If the world was perfect - that would never be a problem.
If you were to accidentally short the output (or display goes bad) - the differential shoots way up to over 200V and the regulator tends to die rather nastily.
Also, per National Semi tech rep - just the act of turning the machine on will eventually kill the parts. The input rises much quicker than the output putting a huge differential between the two momentarily.

One other thing to consider -- LM317AHVT's have just gone obsolete.

So is there any part that can be subbed in that won't be overspec? That's seems like a really bad idea to use the wrong parts for the job.

No, unfortunately.
And Yes to the second statement.

Only sure fire fix to this is to remove gas discharge displays, replace with LED displays and pull fuse to the HV supply.

You mean on this particular board, right? I find it difficult to believe that someone can't design a HV power supply with currently available parts.....

Yes - for this particular board which uses the LM337 and LM317's. I believe the later versions used TL783's for the HV supply. A bit better but still exceeds absolute max ratings.

The original design that Bally/Stern used was a better design, they were "OK" but not that great because they cheaped out and didn't use any current limiting - relied on the output fuse. That works but not the best choice.
Not sure if anybody makes replacement SDB boards with decent HV regulators at this time.

Consider @Barakandl's replacement Solenoid Driver Board who's HV circuit is based on Ballys design with new parts.
https://nvram.weebly.com/new-pcbs.html