After a ride in the ultrasonic cleaner, the parts looked much better.
I found an old stock carb kit on ebay, which worked excellent.
It didn't take long to run into a problem, of course. The original throttle plate assembly is messed up, the secondary is completely seized. I have a second EA61 carb for parts, but it is even rustier. So, I started digging around in my parts. I have this 1300g intake that I wanted to use, where I made my first discovery: the EA61 carb has a smaller bolt pattern than the EA62/63/71 carb.
On the left, EA61, on the right, EA62.
This is the late style gasket over the early manifold, definitely not compatible.
On the left, EA61, on the right, EA71. I have a EA71 here for spares, and for fun, I pulled the carb to see if the throttle plates were any good. They move exactly as they should, thankfully. Even better, the carb base is the same size on both carbs, so I was able to simply bolt the EA71 base onto the EA61 body. The biggest difference is in the throttle plate size, though it's the secondary that is drastically larger. I won't know until I bolt it all up, but I think this will work.
This is the completed franken carb. I did have to swap the linkage parts over onto the new baseplate, but it is all the same size and it was super easy. One more piece to the puzzle!