In switch test mode, do the coin and credit buttons work?

If you pull 1J10 and 1J8 and short 1J10 pins 3, 4, 5, & 6 in turn to 1J8 pin 1 does it add coins or credits?

I don't see the start button anywhere in the schematics, is whatever connector it's using functional?

Something is amiss here. Testing your switch matrix on the board using jumpers did not work, but it does work when the playfield is plugged in?

Perhaps I gave you the wrong connector or pin numbers? http://www.pinwiki.com/wiki/index.php?title=Williams_System_9_-_11#Switch_Matrix_Row_and_Column_Testing
It may help to determine if you have a board problem or a playfield problem. The pinwiki and your manual can help you find out.

Page 6 in the manual also mentions "no credits may be won (and coins are rejected) while the coin-lockout coil is de-energized".

Sounds right. I'm leaning towards a board problem but before you ping Quench GRUMPY cfh or barakandl you might try disconnecting every connector that goes to the playfield and lower cabinet, then using your jumper to add coins. If you still hear the credit sound but don't get credits (try adding several "coins" just in case the game thinks it needs ten to give a credit or something crazy) maybe take a look at U15 and if it's socketed, reseat it?
http://www.pinwiki.com/wiki/index.php?title=Williams_System_9_-_11#System_9_Switch_Matrix_Explained

The confusing part to me is why didn't the factory reset change the coin settings back to default? LTG answer should have fixed it. It shouldn't be possible to do a factory reset without the coin settings going back to factory default.