The board on the 1 pt score reel on Big Strike is used to fire the change relay:
Big Strike 1 pt DU (resized).jpg
The odd numbers fire the relay and a 500 ohm resistor added probably to reduce arcing on the score reel board. If you look closely at your board it looks like every other position has overheated which likely corresponds to the odd numbers. Rather than arcing I suspect that your board got dirty (between the WD40 and all the grime it probably attracted) which increased the resistance between the board and the rotating contacts which generated the heat that toasted the fiberglass.

It's worth a try. If it doesn't work well score reels aren't hard to find.

I doubt the board cooked your coil. It could have been the end of stroke switch on the 1 pt score reel which would have held the 1 pt relay on. How does your 1 pt relay coil look?

Also, for those unfamiliar, Big Strike/Hot Line may be the first use of a dot matrix display in pinball. If you haven't seen it there's an animated GIF on ipdb at: https://www.ipdb.org/files/1245/Williams_1966_Hot_Line_Animated_Rollover_Button_Letter_Sequence.gif. Another excellent Steve Kordek innovation.

That is freakin COOL!
Thank you again Mr Kordek!

Thanks for the gif link!

Is the 500 ohm resistor intact and good?

Bakelite looks burnt up for sure. I don't think it runs high enough voltage to be a problem, but make sure it isn't full of tons of carbon soot making it conductive before coating over it.

Crazy how that burnt up like that.
Best wishes on the repair

The object of these two games is to complete all the letters of HOT LINE. At the start of the game the display shows an H. As you roll over the white buttons the lights below them go out. Once all the lights are out the H light at the bottom of the playfield turns on to show that you've earned it, and then the display shows an O. The process repeats for each letter.

Something is nagging me a little about one of your photos:
Big Strike 1 pt DU2 (resized).jpg
If you follow the copper traces from the contact pads around the spindle out to the edge of the board you can see that the 3, 5, 7 and 9 pads have all been comparably overheated. The number 1 pad however looks just fine. That doesn't make sense. I wonder if there's any continuity between the number 1 pad and the 3, 5, 7, and 9 pads which should all be shorted together to the blue-orange wire.

That happened sometime before the burning started. That means that the Change relay and 500 ohm resistor never drew current through the #1 pad so it never overheated.

It will be good to check the resistor too as dr_nybble suggested. If its value has dropped significantly that could explain or contribute to the overheating too.

Dunno. Do we get a hint?

Can you fine sandpaper those coper pads down so they are totally clean on that disc? Shouldn't have any marks left...if you can.

It could use a bit of a cleaning . Since the coil and resistor are wired in parallel you would need to desolder one leg of the resistor to get an accurate resistance measurement. According to https://www.flippers.com/coil-resistance.html the resistance of the coil is 14.2 ohms which would dominate the measurement of the two in parallel. For reference a 14.2 ohm resistor in parallel with a 500 ohm resistor has an equivalent resistance of 13.8 ohms. Given the variability in actual coil resistance and the resistance between your meter probes and the circuit you couldn't make a meaningful measurement of the resistors's resistance without disconnecting it from the coil.

Your photos don't show any signs of damage so I'd be surprised if you find it's resistance is far from the 500 specified on the schematic.

Awesome! Yeah, I would try cleaning all those contact points and the big ring in middle all pretty before teflon lube.

Make sure points on all the fingers are clean and mating well as well

I own Hot Line, which is the replay version. One of the games of my teen years. You can complete a letter three ways. Rolling over the buttons, going down the inside outlanes, when they are lit (once you get to a certain point you can't do that any longer) or lighting the A/B which opens a return gate to the shooter lane. If the ball goes back into the shooter lane, there's a rollover switch there that will score the current letter and advance the sequence.

Once you complete the I, the outside outlanes light alternately light for special. If you complete the N, it lights another special, and if you complete E you get another replay (or two, it is selectable).

Fabulous game. I restored mine completely, new bgresto backglass on it and it's so much fun.

Here's a neat factoid for anyone who's interested in EM game design. Compare the Change relays on Big Strike (add a ball) and Hot Line (replay):
Big Strike Hot Line Change relay (resized).jpg
Both games toggle the Change relay for each 1pt scored but they do it in different ways. Hot Line drives the Change relay from the Match unit, which is advanced by the 1pt relay, and has no resistor in parallel with the relay coil. Big Strike drives the Change relay from the 1pt Score Reel and adds a resistor in parallel to the Change relay coil. Why the difference?

Big Strike is an add a ball game with no match unit which explains why they had to use the Score Reel instead. The extra resistor is probably there to reduce the arching on the relatively light weight contacts on the Score Reel contact board. The contacts on Hot Line's Match unit stepper are heavier duty in comparison and don't require the extra protection from arcing.

Are any of these relays energized? If not do the same to these switches

Pinball (resized).png

It took me a year of owning the game to realize that it was spelling out Hot Line. Just thought they were random patterns.