I only see a couple of things that score 10 point. (3 stand up sws) and the 50 point relay.
You may also be stuck in a reset where the 10 point zero position is not registering on the score reel. The 1/2 reset relay would be pulsing. (normally in the back box).

Looking at the (Space Mission Schematic)
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Found the schematic here
https://www.ipdb.org/machine.cgi?id=2253

Of your two issue the left Flipper is the easier to trace.

1. Look at the switch itself to make sure it is making good contact (clean with a points file or flex stone). If it is clean and making good contact.
2 Then check for broken wires (Give the wire a gentle tug)
2a. At the switch and at the flipper coil
2b. at each jones plug both male and female side.. There should be two, one from the flipper to the main board and one from the main board to the playfield.
Check both the Yellow and the Black/White wires. (Note they were running out of wires when they made the machine so your colors could be different).

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Free play not working

This one can be trick depending on how they set up free play.

Here is the circuit and the an image of the credit unit.

For free play to work they normally closed the credit Unit Zero bk Switch. If you put credits on the machine by manually turning the credit wheel does the start button work?

I suspect whatever they did for free play stopped working. (Sometimes people use a clip to hold the switch closed and it comes off).
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That looks like the Match unit at 10 step unit (Pages 12 and 13) on the Williams parts book
http://www.planetarypinball.com/mm5/merchant.mvc?Screen=BOOK (Books)
http://www.planetarypinball.com/reference/partsmanuals/WMS_Parts_1977-1978/index.html#/12/ (exact page)

The Alternator switch activated the change relay

Look for the Yellow and Black-red wires on the match unit.
If the change relay is pulling then check the switches on that relay.
Here is a pic of my Argosy match unit.

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