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Parts for GE GSM603L-02 Dishwasher - AppliancePartsPros.com

The missing rubber flap would not effect the drain.
I am assuming that the rubber flap you refer to is Item 557 in Section 03.
You are off a bit on what it does.
It actually closes when the pump is in wash mode. This stops water from gushing out the hole instead of going up into the spray arms. The symptom would be a poor wash or you may not even notice a difference.
Then during drain it opens and lets the water from the bottom of the tub drain into the pump so it can be drained out.

I am assuming that you replaced the pump because of poor draining.
Also that the wash/drain solenoid was also replaced as part of the pump replacement kit.
These solenoids can cause problems re: they can overheat and melt the internal plastic then they either do not open/close or do not pull the solenoid piston all the way down.
If not.
Check that the solenoid pulls all the way down (drain position).
Also remove the spray arm and look down the pump throat to be sure that the wash/drain flapper is closing all the way.

If all OK I would remove the drain hose at the house plumbing and drain it into a pail just to be sure that it is not a plumbing problem.
Be sure to hold the drain tube securely and put a towel etc over the pail so it does not splash all over the place.
It is a good idea to have another person there so they can open the door and stop the drain cycle if there is a problem.

There is also a long shot that this is a unit that does not match the parts breakdown above. Sometimes this can happen when they update a unit re: the model numbers lag the updated parts.
If there is a screen filter at the back/bottom of the tub then this may be the case. If the unit has this remove the screen filter to see if there is a piston and nut (valve) assembly mounted in the tub floor.

I am going to the cottage for a few days so will not be available.
When the forum is busy if other regular posters see that the post is being answered they do not have the time to check the post. So If you require help when I am gone you may want to start a new post with a blurb about what you have done so far and a link to your original post.

I am back from the lake.

The way the drain works is that the timer activates the solenoid for about 5 seconds. This pulls the solenoid's plunger down, flapper valve moves to the drain position. The water pressure then holds the flapper in the drain position till the unit is empty and then the spring/s pull the flapper back to the wash position.

If power to the solenoid is left on too long it will overheat.