Check the solder joints at the flipper relay and the contact in the flipper relay.

Also noticed in the 2nd pic that the bakelite piece that joins to the plunger isn't sitting against the bracket, meaning the spring has lost tension and/or the mechanical movement of the assembly isn't free.

Are you able to pull on the flipper bat from the top of the playfield and notice some slight 'play', movement up and down? This is a must.

Yes the flipper relay is on the solenoid driver board as you indicated.
It's the contacts in the relay, fixed and moving that are of interest, you should clearly hear your relay pulling in and releasing on power up.
I personally have found removing the relay cover to be difficult, usually resulting in breaking one of the latching pieces that are really narrow, and being so old they don't seem to have much 'give'. But it is possible. You will notice on the short faces of the clear cover that there are two small lugs locking into the cover on the base. So to remove, the cover needs to lift over the tabs.
Notice in your second last pic that just shows the relay, it appears to have had the cover removed, or possibly attempted to be removed as the clear cover at the base I mentioned looks to have a crack in it.
But that may not be necessary as you can usually get a good indication of the condition of the contacts through the cover.

Just and observation and suggestion, from looking at the last pic, when reflowing and soldering in new header pins, whilst adding solder and watching it melt/mix in with the original solder, keep your soldering iron tip on the pin slowing lifting the soldering iron tip upwards so as to draw the solder up, and surrounding the pin, not just at the base. The more you do the easier it gets.
I'm not suggesting to add heaps of solder, but you will get the same affect covering the whole pin with minimal solder.
Some of the solder on the pins has taken/been applied well, but look at the ones that 'appear' to not have got heat for long enough and the solder has formed a sort of dome around the pins meaning it hasn't taken to the pin at that particular point.
And again this is just a suggestion and I'm not saying it's a must to do.

the EOS may be opening, but is it actually working?

with one wire removed from the coil, put your meter to the the removed wire and the other EOS switch wire on the coil and are you getting close to zero ohms/continuity with the switch closed?

also what are the windings resistances? the thin wire winding and the thick wire winding.

can you post pics of your wire connections to the flipper coils, showing the diodes?

not that it's related, but looking at the second pic posted, the insulation wire on a white/grey? seems be be wearing away on the flipper bracket?

just for safety i'd undo the philips head screw and adjust that cable gland so the wires are away from the flipper bracket.

hard to tell from the latest pic, but does that EOS switch have good connection between the contacts, and are they tungsten tipped contacts? It could just be the pic but they look to be standard switch contacts? and they don't look to be making very good contact nor in the best shape. I read you tested them, but it only takes the smallest amount of surface area to get a good reading, but these contacts are transferring a fair bit of current and the larger the contact area the better.

to measure the resistance of the windings you need to put a piece of paper between the EOS contacts and remove the far right single orange wire from the coil. Yes you can test with diodes in place, you just need to remove any 'circuit' so the meter is measuring the windings only.

no offence, but maybe remove the wires, strip back some insulation on each, twist together the rh side ones, tin the wires (adding some solder) then re-attach to the coil lugs, for each wire do the same. It will ensure better connection and remove any stray wires.

i did read back a bit about your voltage readings, but what voltage do you have at the flipper coils? about 43VDC?

it doesn't appear to 'snap' in hard.

something i read recently and that's if i got it correct.......that it may be the coil/s, i get it adds up dollar wise but but any other brand than PINCOIL.

and not a big deal, but that gap of the EOS when fully open, ideally wants to be at least 1/8 th of an inch (3.2mm).

it sounds like the flip/hit winding isn't working and it's the hold winding/ like an issue with the winding of the coil/s ? Maybe.

you did put the springs in over the plunger?

just a thought, as your flipper crank is different you have the option of modifying it a little to incorporate the add on of an outer return spring.

one 'pointy' section has the stud to open the EOS, you could use the other side to attach a spring to and a ground tab terminal at the EOS switch mounting?

I have used those same kits with no issue of the flipper bouncing.

you could also try stretching the spring to give more resistance and help with the return, but this 'may' have an adverse effect on the pulling in of the plunger?

did this happen before? you could use your original bakelite links if they are okay?

you did offer to post a video of the bouncing effect, I'd be keen to see it.

interesting videos to watch, thanks for posting.

this may seem ridiculous, but can you try and compress both the springs between your fingers and feel if they are the same, it's not the correct way of doing it but it should give you an idea if they are the same. With so many compression springs available it's unlikely but possible one is softer than the other?

and just for safety sake I'd move the diode away from the LH flipper plate, it looks like it's touching.