You were exactly right. I hadn’t chased the whole wire down because it was wedged behind the starter. I pulled the starter and found that whoever put the starter on last had smashed the wire between it and the mounting point. I removed the smashed wire, spliced it back together and it now works...
I’ve got an old Dixon ZTR mower with a Vanguard 15.5 Briggs and Stratton motor. There is a small black wire that connects to the ignition coil and when it is unplugged the mower will start but not shut off; when it is plugged in the mower will not start. It seems to be that this is the kill...