Your battery is on borrowed time as well. If the regulator was gone, you would overcharge the system. Sounds like the diodes in the alternator were fried. You shut the car off at night, all is good. Start it the next morning, no go.
Diodes allow system charge to move one way, Alt to battery.
The battery was slowly draining over night. After discharging the battery many times in winter, it is probably done as well.