I have a very similar 1953 United "Chief" bowler with very similar problems when I got it.
The trick is to clean and lube every moving part. The reset bracket that held the coil was broken, so I repaired that which helped with resets. You will need a burnishing file go go thru all of the contacts on the steppers and the puck contact switches. Steppers are critical that they snap to the neutral position when the reset coil is pulsed. I used special grease on the stepper contacts - made for stepper contact. The more you reduce friction, the better it will play. And all the moving arms, use synthetic grease and synthetic oil ONLY. Lithium will eventually harden like the old petroleum based grease. Synthetic will stay soft forever.

After going thru everything twice, some three times, the machine works flawlessly now.

Fixed.

Here's a link to a high resolution of the schematic on my Chief. I'd bet is virtually identical to your machine:

http://www.ramegoom.com/john/United_Chief_Shuffle_Alley/schematic01.jpg

I always use this stuff on all of my machines' stepper contacts. It's special grease that doesn't turn into carbon when the contacts arc, and makes your stepper contacts last longer. Much better choice over petroleum-based grease:

pbr-grease (resized).jpg

Mopar - Looked like the one on the left, maybe?
img02 (resized).jpg

This one is a 1953. The pins are shorter than the standard pins, 8-1/2 in. long, and have a special mounting - read that "impossible to find".
pin (resized).jpg