The common element between the start failures and the lack of a game over is the "On Ball Release" switches. If those two switches are not closing, you'll see both those issues. Check the switches attached to the ball gate in the ball trough. When that gate falls (after points are scored on ball 1), both switches should close - which then enables end of game detection and the replay switch.

BTW, correct operation is - after points are scored on ball 1, pressing the replay switch will start a new game.

Yes, there are three switches in that stack - a NC switch which enables the Replay button circuit, a NC switch that enables the triggering of the "O" (Match) relay (which then triggers the Game over relay) and a NO switch which is the hold switch for the ball release coil.

Is the ball gate closing far enough to close the two NC switches? With the game unplugged, use a meter to verify conductivity when closed.