I assume the reset works fine during the self test. As Steve says above make sure you know and are testing the sequence properly.
Also make sure the switches on the back of the target assembly are each working. You may have a bad target down switch.

Looks like the target reset is controlled by switch 23 on the MPU. ON-no reset, off-reset after deluxe.
Keep in mind if you change a switch you have to reset the game to take effect.

The reset coil(s) are controlled by the solenoid expander and require a diode in series as well as across the coil. That's why there are 3 lugs.
Normal wiring for this game is shown on the schematic but since the coil fires in self test it should fine during game mode. Perhaps switch 23 is dirty or bad? or the solenoid expander doesn't operate properly. When you put the game in lamp test do you hear the sol expander clicking? Sounds like a cars turn signal clicking.
Try turning the game off and switching 23 back and forth a couple times to possibly clean any internal corrosion.
There are only 2 possible behaviors of the drops and targets as shown below.

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No matter how switch 23 is configured the targets should always reset between players on a multi player game.
Resetting the targets requires the sol expander activate. What you want to do is verify the expander is in fact activating between players. You can do this by observing the dummy lamp under the playfield that helps activate the expander. After the ball drains the game should light that bulb indicating the SCR is attempting to energize the relay, you should hear the relay click at the same time and then the reset coil should pull.

One thing that comes to mind is replacing the expander bulb. You will want to do that with a new 555 lamp. Some times the 555 gets removed to do a quick swap on another lamp, other times it has failed, and it might even have been inadvertently been swapped out with an LED.