The small screws are just for the coil windings - the control/trigger portion. If, when you said the "motherboard is burned up", you meant the dark patch (post #5, third pic) on that green fiberboard (which is just a HD insulating spacer), then that heat came from the high-current contacts having too much resistance. But they don't look discolored or dirty in your pics. Are those before cleaning, or after? The critical points are where the silver "T" bars (the slugs & moving contacts) touch the copper bars (the stationary contacts).
I assumed the coils still work. If you momentarily apply 12V across each coil, will it snatch a slug inside the tube? Don't try to hold the "T" while you do that - it could pinch... And if the spring on the "T" is stout, be careful where it's pointed; when you break the circuit, it could shoot the "T" out.
Once it's clean & tested, it needs silicone grease (dielectric OR electric) on the copper shafts where they pass thru the case, on the case seam, under the small screws, and any other case penetration to make it water-resistant. And it needs electrical grease on the 16 contact points (4 on each "T" & the mating copper pads) to keep Oxygen, CO2, ozone, & Carbon away from the metal when it's hot. Avoid greasing any threads, but don't worry if there's some on them. Clean & electrical-grease the upper face (AWAY from the case) of each nut on the copper studs so it makes good contact with the heavy ring terminals.
At that point, it should be better than new, since it doesn't look like they used any grease. When you mount it, try to set it so all the case penetrations are LOW, and the airtight/watertight part of the case is HIGH so it acts like an umbrella/diving bell and drains water instead of collecting it.
You sure your not just messing with me Steve?
I might have messed with you about just general :bsing in the past, but never about something technical. And this particular item is genuinely interesting to me. I've never come across a relay like this, and I want to see it working again.

opc1: I'm not guaranteeing that it'll be reliable enough for you to put back on the winch permanently, but it's really a quick/cheap process just to clean it up & check it out. If it doesn't work, we'll each be out an hour or so, but we'll both have learned about this relay. And you'll still be able to replace it with "normal" winch relays if you want. But I fully expect it to work at least as well as new with a few more minutes' effort.
That "someone" is Badlands winch manuf. and yea... it's likely as cheap as they could find.
Badlands is just a sticker that goes on junk shipped into the US. And it's one of several that
all come from the same assembly line in China. There are probably several factories buying those relays for winches & other 12VDC motors, but I doubt it was designed or built in-house by the same factory that's putting it on these cheapo winches.
Here's another one I haven't encountered before:
http://www.ebay.com/itm/130839390780
In any case: it's electrically equivalent to the relays on high-quality winches, so if you want that old winch to work, and this relay doesn't, you can certainly buy a pair of common replacement relays (probably also made in China) and slap them on there.
Here's the style my Ramsey uses:
http://www.ebay.com/itm/371145803325
Here's an SPST continuous-duty relay:
http://www.ebay.com/itm/281154578706
It would take 4 of those to replace your original 1 polarity-reversing, or my original 2 DPSTs.