Many of the very oldest ones, yes.

Gottlieb didn't start putting game over relays into their games until 1957, so the flippers stay working on their games before that. There are some exceptions however, such as the turret shooter games from around 1950, and even the first flipper games like Humpty Dumpty have a timer that turns the flippers off eventually.

Have 2 Gottlieb Woodrails, yes, the flippers are always on, when the game is also on.

The last woodrail I restored was a Gtb Auto Race. No game over relay in that so the flippers were always on, though the backglass did show a game over pane on the backglass, yet no lamp behind it. So it was obviously in the pipeline but not used on that one for some reason.

I agree with the previous posters that there would not be a Game Over relay unless the circuitry demanded it. Any game that recycles a single ball (turret-shooter bottom, multi-player games, add-a-balls, etc.) probably needs one. Any game with an end-of-game match needs one.
.................David Marston

I happened to stumble across this recently:

"PM: This was the first single-player game with a match feature. Who came up with the idea for the match feature?
Wayne Neyens: I have no idea. I think we already used it on multiple-player games. All our multiple-player games had a match feature. Now we finally got some sense in our heads and killed the flipper when the game was done."

From the extended interview with Wayne Neyens in Pinball Magazine, issue 5, page 150, talking about Royal Flush from 1957.

/Mark