This afternoon, I prepped all of the boards for my new parts. I hope they will arrive tomorrow since I paid for priority shipping. All of the afflicted components have been removed and the boards have been cleaned and fresh solder blobs have been applied to each of the pads. Have a look:
Above, you can see the caps having been removed, revealing much goo. Then I cleaned it. Many of the caps are also gooey and crusted.

Three caps beside the RAM sockets all removed and clean.


Partial way through and complete. Here is all that I removed from the motherboard. I believe that the LM3080 is perfectly fine! All of the legs cleaned up and are now coated in nothing but shiny solder.

Sadly my Apple IIe card lost a pad, but in my defense, it was the first SMT cap I removed.
Looking back on the experience, one pad lost out of a hundred is a success! I will simply have to wire the second leg to the bottom ground plane like it was originally.