So, I went a little deeper into my electrical problems tonight. Following
cheesehead's thread, I'm 99% certain that the issue is with the rotor. My stator looked good when I tested it with the ohm meter and the battery is new, so I think I can rule that out. When I checked the rotor with the ohm meter, I got absolutely nothing.
However, I did notice a couple things when I took off the stator cover. First, I saw some small bits of cable shielding fall out. And second, I noticed that the copper wire from the coil had a loose end. Could either of these be an issue?
Whatever the case, looks like I need a new rotor. I saw a bunch on ebay for 70's cb's, but I am not sure how compatible they are. Any buying tips or places I should check out?