Only connection the headlight switch has to the ignition system is the factory splice with the black alternator, black ammeter, and red ignition switch wire.
Honestly thought you were pulling everyone's leg.
Anyway.
1) if that's the only connections at the main splice, it's missing one. Should be also be a hot to the fusebox. That's the way most are but check the factory service manual..
2) none of that is related to the ignition or the starter relay. That's the main junction. It's always hot. Power goes there onto whatever is calling for it.
As others have said, there's a short between the headlight or marker lights or instrument lights wire and the start wires, usually lablled S in the factory diagrams.
shown schematically here
