Hi mrbanjo +
in Europe we can buy new FINDER relays of all kind - simple relays and ministeppers (1 -> 2 -> 3 -> 4 - > 1 -> 2 ...) or relays with "built-in delay" (delayed pulling in etc.) - I LOVE the FINDER relays.
Here the link for an simple 24V Alternate Current Relay with 2 Make-and-Brake Switches (we could use in Gottlieb and Williams pins): http://www.tme.eu/gb/details/40.52.8.024.000/miniature-electromagnetic-relays/finder/405280240000/ --- I click below of "picture of the relay" on "Specification" -> http://www.tme.eu/gb/Document/29c8f5122befaa693b8c4b815543fabe/s40en.pdf -> on the first page - on top, outmost-right -> the relay is now of color blue. I scroll down to page-7 -> down on the bottom, on the left -> the "24 VAC version of this type of Relay" runs well on 19.2 VAC to 26.4 VAC.
On top of page-7 is a list for "Direct-Current-Type of this kind of relays" -> the 9.024-coil-code "24VDC" relay runs well from 17.5 Volt DC to 36.0 Volt DC (((the 9.036-coil-code "36VDC" would fit in the Chicago-Coin pin - it runs from 26.3 VDV to 54.0 VDC)))

From the fact "FINDER" produces 24VAC-Relays AND 24VDC-Relays I read: When we have a DC-Relay -> we MUST have Direct-Current - the 24VDC-relay does not work on 24VAC.
The picture in post-41 shows the information "Sigma 67R4 24 DC" -> THIS Relay NEEDS DC. I think I read: contact-points on the switches can handle well up to 120 VAC, can handle well up to 3 Amperes. I also see the drawing of four Make-and-Brake-Switches.

mrbanjo, Yes -> post-42 -> JPG -> encircled-red: The diode (used on all the problematic relays) - the diode in question. I believe: When this doide is "broken" and let current flow in both directions -> the Specific Sigma-DC-Relay gets AC-Current and acts faulty.
Post-26, JPG lower part -> schema-snippet-A-1 -> a diode for the several 100'000 points relays.
I suggested to make more than 100'000 pints -> does the relay pull-in and stay pulling ?
IF YES -> diode is GOOD -> diode is "5612-4004" --- the same diode is used on the problematic A-, B- etc. relays -> want to try the GOOD diode in place of "diode in problematic-area" ?

I am not good in electronics - I cannot help in testing diodes and capacitors ...

(((If You (beeing in the USA) cannot reach the site and the specifications (PDF) http://www.tme.eu/gb/details/40.52.8.024.000/miniature-electromagnetic-relays/finder/405280240000/ --- write here in pinside -> I can show stuff by making snippets using the "PrintScreen" button. Greetings Rolf

P.S.: About "new" relais: The disadvantage on FINDER-Relays: They only (mostly) have only TWO (Make-and-Brake) Switches --- when I need 6 switches -> I must mount three Finder-Relays PARALLEL (coils mounted parallel). Maybe in the US "easy to buy (?)" are OMRON-Relays: https://www.omron.com/ecb/products/pry/ -> General Purpose Relays: http://www.ia.omron.com/products/category/relays/general-purpose-relays/index.html -> for Control Panel: http://www.ia.omron.com/products/category/relays/general-purpose-relays/for-control-panel/index.html -> Miniature Power Relays: http://www.ia.omron.com/products/family/948/ -> Line-Up: http://www.ia.omron.com/products/family/948/lineup.html

mrbanjo, when "everything else" does not solve the problem -> want to buy NEW AC-Relays of 36 VAC, get rid of the diodes -> mount the new relays and do make new connecting-wires to the new relays (I believe: ALL new relays come with "Make-and-Brake" switches - we easily can use two of the three blades of an M+B-Switch to use for an Normally-Open-Switch / for an Normally-closed-Switch).

Looks like the diode is bad. Can you check all the other diodes on that board also to compare? Diodes typically will show a measurement of 0.1v to 0.8v depending on what kind of diode it is. So in one direction(leads one way) it should read 0.1 to 0.8v and in the other direction(leads the other way) it should not read anything. This is because diodes only conduct in one direction. The diode is converting the AC into DC and the Capacitor is there to filter/smooth out that voltage. So for the diode to conduct in one direction the Negative lead of the meter would go on the side with the silver band and the Positive lead would go on the other(obviously) . Different DMM (Digital Multimeter)'s might be slightly different in their readings but the a diode should still conduct in only one direction.

DiodeForward_(resized).JPG

Diode_Theory_(resized).jpg

Diode_half_wave_(resized).jpg

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Looks like it.

Not sure.

That would do it.

1N4004

Also, you could measure the 3000's relay diode and the Change relay diode since you said those both work and those are used the same way.

You're very welcome! Enjoy!