Voltage regulator smoking?

I too question whether or not the regulator itself failed; but lets call that possibility one.
We see some wiring modifications in your photos, so lets also see some photos of the alternator - both sides.

I'm sorry this happened to you. Unfortunately there's no clue on the ammeter that when the needle is near either extreme (or beyond) this is a critical situation and needs immediate action.

As mentioned above, the ammeter is located in the battery feed/discharge line.
The needle is moved by electric current flowing past it. Or more specifically the small magnetic field created by that flowing current.
When current is flowing into the battery, the needle points to C for Charging.
When current is flowing out of the battery, the needle points to D for discharging.
The ammeter is calibrated to show 40 amps to 40 amps. The middle (zero) is normal after the battery is finished charging.
Anything over 20 amps for more than a few minutes is going to be a problem for connections, wires, and sometimes the ammeter itself depending on model and condition.

Fortunately the circuit is pretty easy to follow on your '66. Of course we have to figure what the previous owner(s) modified, but the basic circuit is pretty simple.

Now the bad news. When there is no votlage regulation all electric circuits there were on while you were driving saw much higher voltage. Ever notice that lights are brighter when the engine is running? That's because with the engine off they get electricity at 12 volts from the battery. But when the engine is running they get power at 14 Volts from the alternator. So if any lights were on and the alternator was supplying power at 16 or 17 Volts, might have burned them out and/or fuses and connectors may have got hot. So you will want to check all circuit connctions and lamps that were on at the time for signs of heating. This includes the ignition.

Votlage regulator: We see a mechanical regulator in the photos. Take it off, open the cover and take some photos and examine it. if the points are stuck or burned together thats likely the cause. if not, then we need to do some more investigating.