I can check my schematic when I get home – but it will be late tonight.

On my Solar City, (if this helps), the 10 point relay 'N' shows that there are nine 10 point contacts on the playfield, including the Kicking Rubbers switches.

Also consider this possibility:
https://pinside.com/pinball/forum/topic/gottlieb-em-tech-tip-mystery-stuck-switches

I thought the issue was with 10 point relay staying energized? Is the score motor (on the bottom board) also running? Don't quite follow (and not sure what you mean by the switch being 'energized').

A picture of what you're seeing might be helpful too, if possible.

Check your eos switch in the score reel. It sounds like it is sticking on.

By "score motor" do you mean the "score reel"? (The score motor is the metal circular assembly in the bottom of the cabinet; the score reel is the part in the backbox that displays the digits for the score.)

Guessing you mean the score reel.

When the N relay activates, a switch there closes and energizes the player's 10's score reel via a separate switch on the Player Unit (there's one switch for each player). This will pull the plunger in at the score reel.

But then the end of stroke (EOS) switch on the score reel needs to open, to cut power to the N relay.

So either the signal isn't reaching the score reel to energize it, or the EOS switch on the score reel isn't opening to turn N back off. Either one will leave N stuck energized.

Which Jones plug is disconnected? The one that comes from the playfield? If that prevents the N relay from being stuck on, then that suggests there's a 10pt scoring feature stuck on the playfield.

But for the other issue, when you manually activate the N relay, is the M relay also activating?

The ball count function is integrated into the Player Unit, which is the large stepper unit located about in the middle of the backbox.
Some simple things to check are that the 1st Ball relay (DX) is latched, and that the P relay energizes when the ball crosses over the ball trough switch. But there are around eight switches in the path to the Player Unit that need to be working in order to step it to the next player/ball, so it can be a bit complex to debug sometimes.

Just turning the game on should light Game Over and Tilt, without pressing the replay button on the coin door.

Then when the replay button is pressed (and there are credits), the Start relay should energize and start the score motor running.

The 1st Ball relay (DX) should latch, and this should complete a path to energize and latch the AX relay (Reset Control). Note that both DX and AX are 'interlock' style relays, so only energize long enough to physically latch the armature plates, until they are reset and unlatched (vix DXR and AXR, respectively).

So sounds like the first thing to debug is getting S to energize, have it start running the score motor, and then get AX to energize and latch. Then from there, the score reels should reset, and the Player Unit needs to reset to complete the sequence.

The complete startup sequence can be found here:
http://user.xmission.com/~daina/tips/pub/tip0457.html

If you manually set all the score reels to 0 (so they don't need to be reset), then does the score motor stop running? (Just to confirm that's why the score motor is running - to reset the score reels.)

N and M shouldn't be energizing during reset though. Are they activating during reset, or are you saying that's a separate issue?

M will energize whenever a 10's score reel is rolling over from 9 to 0, and the N relay is active. A 9th position switch and a "on add tens" switch on whichever score reel is carrying over both close, completing the path to energize M.

So I'd guess one of the 10's score reels is stuck trying to carry over from 9 to 0.

To be accurate, it's not actually in the set of score motor switches. The reset is controlled by the AX relay, and a switch closed on AX when AX latches causes the score motor to run until the score reels are all reset to 0. Then the AXR coil energizes to unlatch AX, and the score motor stops running. AXR energizes as a result of all the score reels reaching 0, but not directly; the Player Unit has to finish stepping to its reset position as well, which also requires all the score reels to have reached 0.

All of that is controlled by the Player Unit advancing. After the ball drains and gets kicked out over to the plunger, it passes over the ball trough switch (which is under the playfield), and that switch closing should activate the P relay. The P relay energizing should cause the score motor to run, which generates pulses to advance the Player Unit the correct number of times, depending on the number of players.

So the first observation is whether the P relay is energizing when the Ball Trough switch closes.

Activating P, and getting the Player Unit to advance, both involve several switches to be closed in their respective paths. Using the schematic to check all the associated switches along those paths would then be the next thing to do.

Yep.

If P is energizing, then that means it's getting the signal from the ball crossing the ball trough and closing the switch there. The score motor should run, and the Player Unit should advance from the switch at motor 1A closing. So the signal is getting lost somewhere along the logic patch of the switches that feed the "Add Player Unit" coil. This is section 15E through 15I on the schematic.

On the Credit Unit switch stack, the middle switch detects whether there is a credit or not. When the Credit Unit is at 0, the post on the cam lifts that switch stack such that the bottom and middle switches open. Can either bend the middle switch so that it doesn't open, or install a small jumper wire to close it, or other similar means to keep the middle switch closed. Then it will operate on free play (and still work on coins too).

The path for this is around 12E-12I on the schematic.

The Last Ball relay (BX) has to have latched, and when the last ball drains and the O relay energizes, the logic first checks for a Match for each player via the wiper disc on the Coin Unit, then after the Coin Unit completes stepping down back to zero, the path to the Game Over (Q) relay completes and Q energizes and holds.

So make sure BX is energizing and latching first, then check the rest.