(A) Solution to tension release not working
Someone told me to take apart the tension assembly on the front and look at the piece that has the -|+ indicated on it. It has a horizontal bar across the middle that goes down the "split" in the tension stud. Mine was flat - and it should work when it is flat. But they said to put it down on a flat surface and use a wide-blade screwdriver to push that bar back a bit (without creasing the part) and try that. It worked!
They explained that as that part is made of aluminium it is reasonably soft and over time can get bent "forward" by the pin, meaning it doesn't push back on the pin any more. I'd call this either very fine-tuning or a workaround - I'm thinking maybe Singer should have put a dimple on the back of that bar - but regardless of what you call it the machine now sews very nicely and consistently releases the thread tension so I can remove the sewing from the machine.
I thought I'd share that solution because it is bound to affect other people sometimes, and it is such a very easy fix.
I'm still interested to hear of any other solutions as, while this works, something else might have been off? The reason I suspect that is the horizontal bar in my tension assembly had not been deformed by the pin over time as far as I could see or feel...