The alternator must not be wired to the starter motor on that car.
More recent cars, maybe around 1980 onwards but that depends on the car, have starter motors with the solenoid built in. Those starter motors have a permanent feed on the big wire and a lower current control feed on a smaller wire. The permanent feed is a good place to connect to if you want permanent power, which is why many alternators are connected there.
Your car, like most of its age, has a starter motor with one wire. When power is applied to that wire, it turns the engine. That has no power on it most of the time and if you wire the alternator to it, the battery won't charge. The solenoid on your car is separate and it is the B+ side of the solenoid where there is permanent power, so that is where the alternator can be connected.
You can remove the regulator. Ford used the B terminal on that to connect the wire from the battery (that comes via the solenoid terminal) to the feeds to the horns etc and the "firewall plug". You need to leave those wires connected together and insulated from ground.
You could connect the alternator to that point if it is more convenient.