This is the normal power up procedure for this era of machine - connecting the power cord and then pressing the flipper button or closing the coin switch will energize the lock or hold relay and distribute power to the circuits.

Placing the ball in one of the playfield eject holes and having it ejected from the hole is normal when the machine is in game over mode.

What happens to machine when it's powered on and the ball is placed in the outhole below the flippers? What happens when you press the red credit/start button on the front of the cabinet?

What happens when you manually push in the coin relay or replay relay?

If the machine powers off after the ball is ejected from one of the four kickout holes, then there might be a burnt coil/solenoid on one of the kickout hole units or on a related circuit. Check for any burnt or damaged coils/solenoids throughout the machine.

You can always look at a mid-1960s Williams 1-player schematic like River Boat , Soccer , Full House , Hot Line , etc. The non-playfield circuity is similar and it might give you some clues to solving some of the problems.

https://www.ipdb.org/files/1965/Williams_1964_River_Boat_Schematic_Diagram_continuous.pdf

https://www.ipdb.org/files/2232/soccer_schematic.pdf

https://www.ipdb.org/files/961/Williams_1966_Full_House_Schematic_submitted_by_Steve_Corley.pdf

https://www.ipdb.org/files/1245/Williams_1966_Hot_Line_Schematic_Diagram_continuous.pdf