Magnavox Odyssey more or less required the screen overlay to play the actual game. Because the system did not contain any real CPU or logic chip nor real graphics as we know them today. Take for example their Skiing game. Forget the ability for the system to create the path, all the system could create is the person (which was a pong paddle). The path of the ski course was printed on the overlay. Without the overlay the screen will just be black except for the paddle.
Vectrex games did not require screen overlays in order to play the game. For the vectrex games the overlays were more to enhance the player's experience. The system was capable of displaying all the graphical elements to play the game but everything was only one color, white. The overlay provided color to games. Pole Position is good example of this 'enhancement'. The system was able to display the car,the road, the landscape in the distance, the other cars on the road you get the idea. With the screen overlay, the landscape was had a blue skyline with green mountains. The road was a brown dirt like look to it. Also some Vectrex overlays at the bottom displayed a quick guide of what the controller buttons do. you can see from the image that button 1 is 1st Gear, 2 is 2nd Gear and 3 & 4 are used for gas.
Arcade games in the 1970s also used screen overlays to an extent, to simulate color graphics for a monochrome game. The most famous example is Space Invaders, which used a color overlay over its monochrome graphics to simulate color graphics.