After carefully testing voltages from the controllers, the original and the one from eBay, the original controller either puts out 12 volts or very low volts at the socket to the regulator.
I think in my tests of the various components in the various configurations possible I omitted to include the eBay controller. It produces the correct voltages at the various settings for the fan speed.
There are at least two parts that are faulty, the original controller that caused the intermittent fan operating, and the original regulator which may have failed when the fan started to run at full speed only when I was starting to look into the problem. The original controller puts out a constant 12 volts now, maybe when it suddenly went from working intermittently to 12 volts it blew the regulator, who knows. I think the regulator normally sees about 7.3 volts max.
The cheap regulator may have only been tested using the original controller or it was NFG when I got it some years ago.
The BCM I got seems to be fine and I will leave it in the car. I shall order a a Behr regulator which is the brand my original one is.
I will use the eBay controller and the Behr regulator when it comes as the regulator requires the antics of a gymnast to install.
I will report back.