Regarding your alligator clip leads, pull back the rubber sheathing and check that the wire is actually soldered properly to the clip, often they are soldered poorly.

Check the rear side of the solenoid driver board.
Look for dry solder joins on the header pins. You are looking for a 'crack/s' in a circular area in the solder where the pin/s comes through the board.

Check the plugs themselves, J1 & J2. A pin could be broken or have poor connection.

I read you are going the simple way first, but IMO those flipper EOS switches should be replaced.

Start a game and using your multi meter put the black lead on any metal part of the body, that being the ground, and then put the red lead of your multi meter on the coloured wire of the LH coil being Green, you should get a reading of about 43VDC.

You can check using the same method on your working RH coil, but put your red probe on the Orange wire.

You will notice the flipper coils, LH & RH have a common coloured wire, being Brown, this is power from the Rectifier Board, the other coloured wires on the coils are from the solenoid driver board.
The wires go from the cabinet to the bottom header/plug of the solenoid driver board, J2 then out through J1 on the solenoid driver board to the playfield.
The relay is in the middle so to speak.
With the flipper enable relay 'ON' one part of the circuit is complete, the other part of the circuit is 'closing' the flipper contacts in the cabinet which provides a path to ground.

I get being new to this it can all be a bit daunting, hang in there, you will sort it out.

Also, IMO paper/cardboard is only good for cleaning gold plated contacts, you should really use at least some sandpaper, if not a nail file/points file or similar to clean tungsten tipped contacts.
And the normally closed (NC) EOS switches with the tungsten contacts need to make very good contact, meaning zero ohms resistance between/across them.

You can remove the green wired pin from the housing, it may be broken/fractured or has lost its tension....pressure to push against the header pin on the SD board.

You can see one side of the plug is solid, the other side shows the locking tab of the pin.
You can push inwards on the locking tab to release the wire and pin, this way you can examine it.
You can re-tension the pin by putting a small screwdriver inside the pin and rotating the screwdriver which 'opens' the pin so to speak, giving it back it's spring.
You will also need to open back the locking tab, possibly using a needle.
But this IMO is a temporary fix, new pins are the way to go, and trifurcon 3.96mm pitch. https://www.greatplainselectronics.com/search.asp
Ed at GPE has what you will need, assuming there isn't a bigger problem, one where he may not stock what you need.

and this is just a personal thing, not saying you or anyone must do it, but when checking a connector I put the multimeter probe in from where the wire joins to the pin, this way you aren't pushing the pin back any more than it needs to be.

and yes i get that a mutlimeter probe is about the same diameter as a header pin, each to their own.

if you find holding the probe in from the back awkward you can insert a needle/nail/or similar that makes connection to the pin then attach an alligator clip to it.

The pin breaking shows the tension was poor, it needed replacing and was only going to be a problem.

you can still test the flippers but bypassing the flipper relay, another issue may be present.

don't stress over crimping, yes the initial expense might suck but once you do a couple of test crimps, your set

so with J1 disconnected from the SD board, and the pins removed from the connector, remove J2 also, underneath the relay.

in essence you are joining the wires/pins together, like for like, so try the green wire (40) from J1 - 8 connected to it's mate in J2 - 2 (20) blue wire.

so you are doing this with the plugs 'disconnected' from the SD board.

i suggest turning the game off whilst setting it up to test.

join the 2 pins and press the flipper button......what happens? It should work.

this same principal should also work for the right flipper, although you know it works. J1 - 9 (70) orange to J2 - 1 (10) red.

the reddish stuff is the flux, it is in solder to help the solder take to the wire and the terminal.

if your in the mood......swap over the coils......or

you could remove the the flipper wire from the RH coil and using an alligator clip join it to the LH coil lug and pressing the RH button see if the LH coil works, if not i reckon the LH coil is a dud.

The way to test a coil's resistance is without any wires connected to it or you will get false readings as electricity takes the path of least resistance.

Have you checked https://www.flippers.com/coil-resistance.html ?

your first pic is where the EOS would normally be attached to, and as in your pic with the EOS disconnected should be reading about 350, so looks ok.

the second pic should be reading about 3.3 ohms.
this is the thicker wire winding.
you could try scraping away some of the coating on the winding before it connects to the solder lugs to check it's the same resistance, if it's lower then there is a poor connection from the winding to the solder lugs.

the third pic should read the same as the first, a little higher, (305+3.3) which it doesn't, which points to a problem, but with the EOS connected it should read about 3.3 ohms.

you don't 'have' to cut the diode off, you can with the soldering iron unravel the diode legs from the solder lug.

i believe you are wanting to get to the root of why the coil is failing?

it looks like you have found a problem which would explain why the flipper coil is not energising.

you may find upon removing the coil that the coil stop is mushroomed or just needs replacing.

if you have the funds a cheap alternative for a complete replacement is https://www.marcospecialties.com/pinball-parts/ASE-1587-101

and modifying https://www.marcospecialties.com/pinball-parts/ASW-A20-34

a razor blade or hobby knife makes for separating the switch stacks easy