I also installed an am/fm radio with dials that I picked up from JC Whitney, a cheap one, not digital and no clock in it. After this work my battery went dead after running the car a couple days before. What do you mean by a worn spot to the #3 wire?
If the problem of dead battery came up after installing radio, then maybe the installation is faulty.
A worn spot would be the insulation rubbing off between two wires. Common problem to 40-year old wiring, if stll original.
You had said in an earlier post, while monitoring current flow, you unplugged the two-wire connector from the alt and current flow went to zero. Good troubleshooting to this point.
Plug in the alt and unplug the voltage regulator, if current goes to zero, then ignition side of vr is supplying coil voltage for the field relay and it is energized allowing current to flow through the alt field. Ignition switch should turn off current flow through vr to alt, unless there is an alternate path.
Now you suggest the problem happened after installing the radio. Cheap radio, so I don't think this would be the problem, but the wiring of power to the radio might be the source of the extra current path not needed. Only takes a few minutes to check the wiring.
Think back in history before the problem appeared. What has changed since then? Retrace the steps and disconnect items added, check wiring at horn relay, unhook the new horn button, etc.