Here is the Schematic and manual (If you don't have one)
https://www.ipdb.org/machine.cgi?id=2196
Schematic
https://www.ipdb.org/files/2196/Bally_1974_Sky_Kings_Schematic_Diagram_continuous.pdf
Manual
https://www.ipdb.org/files/2196/Bally_1974_Sky_Kings_Installation_and_General_Game_Operation_Instructions.pdf

I would check to see if the Delay relay normally closed switch is making good contact (Black/black-Red) wires.
Also make sure the jones plug from the door is seated correctly and fully seated.
I am assuming your are looking at the 1st coin chute. (check the wire colors).
Make sure the 1st chute switch is clean and has the correct gap. and the coin wire is moving freely.

Here is the schematic I am looking at.

pasted_image (resized).png

Looking at the schematic ArgosySK posted above, The 3rd Coin Chute switch on the door should activate the 3rd Coin Chute relay.
https://pinside.com/pinball/forum/topic/bally-sky-kings-problem#post-6474627

3rd Credit chute has only the 2nd coin chute needs to be closed.
these is a NO/NC or Make and Break set of switches on both the 2nd and 3rd coin chutes.

The credit button has a fair number of switches in the circuit

Does the motor run or does the 3rd coin chute relay pull?

CaptureSK01 (resized).JPG

The coin relay only pulls if the 2nd chute is set up for 1 credit per coin.

I believe the 3rd chute just add credits to the credit unit. (And 2nd chute if set up for 2 credits)

You then have to press the start button to redeem the credit.

If the Credit button doesn't activate the Credit relay,
Inspect and diagnose the circuit that ArgosySK posted in post 6 above with Alligator clip jumper wires

https://pinside.com/pinball/forum/topic/gottlieb-big-shot-repair#post-6305153
http://www.planetimming.com/Pinball/troubleshooting/EM%20Troubleshooting.pdf
http://www.pinrepair.com/em/index3.htm#features
Example of a pinsider actually doing this https://pinside.com/pinball/forum/topic/red-baron-tech-question#post-5858156

Wiring at one of the coin chute switches is what got me on this one. See this post.

https://pinside.com/pinball/forum/topic/bally-em-bon-voyage-wont-start#post-2541170

So the chute 3 should have 3 wres
White
Red White (or Yellow)
White red.

So when the a switch is close you should have continuity from red-white to White-Red (Coin drop)
Otherwise you should have continuity from red-white to White.

If chute 3 has a Yellow wire then I suspect your chute 3 is wired as chute 2 and you should follow the schematic for Chute 2.

Sometimes the operate leave all the wires in the back and cap the wires for the chute that is not used.

If so check the caps to see if the Yellow and Red White wires are connected.(The Yellow black for Chute 2 would not be used).

NOTE: they bypassed chute 3 then you would have

Yellow
Red-White (or White)
Yellow Black going to the chute switch

Then check for a capped wire where Red-white and White are connected.

Sorry in text it looks confusing but the schematic shows the switches on the bottom left of my picture. (above)

It would be typical to have just the 2 coin chutes, and different amounts of coin chute relays. For these Ballys, it can get confusing because whatever is shown in the schematic concerning the area of coin chutes may not match up with your specific machine. In fact floating around somewhere there is another published set of schematic excerpts showing different coin chute wiring that might be installed on any machine. I suppose when they were originally ordered from the factory you had options to set up the coin chutes the way you liked. And, they may have been modified over the years.

Unfortunately this means things can get confusing if you are troubleshooting that area because you are trying to understand the schematic to figure things out, but your wiring may not match up with the schematic. You may have to gut it out and figure out all the coin door wiring on your machine so you can reason things out. So if you find that wire colors you are looking for don’t match up with the schematic you might be in that situation.

An example of this is discussed in more detail here in this post 21 in my Bon Voyage thread.

https://pinside.com/pinball/forum/topic/em-schematic-fully-described-from-beginning-to-end-bally-bon-voyage#post-3909686

Here the circuit for the outhole relay.
The Green Starts the process.
Yellow set is while the outhole relay is held in place while the motor runs.

Note: The bonus score should count down 1st before the outhole will trigger.

The next is the process for kicking the ball out.
CaptureSK03 (resized).JPGCaptureSK04 (resized).JPG

Argosy's info looks good.

I'll just not mention the irony that your handle is the same as the problem