Quote:
The "key" to this seems to be the fact that it is necessary to turn the key to the starter position to allow it to run.
Some cars have a by-pass from the input side of starter solenoid [from the ignition switch] to the output side of the ignition resistor, to bypass it during starting. If the current flow from the ignition switch, to the resistor is interupted [dud swirch, broken wire etc.] then the engine will start while the starter is engaged, but will not continue to run when the switch is released to the "ign" position.

Is this what is happening to you?

Replacing the points may well be a good idea if they are worn, but i suggest that they have nothing to do with this problem you have, & how one links the starter position of the switch to the ignition points defeats me entirely.

Have a real good look at the ignition switch. Use the workshop manual to allow you to test the various circuits through it. If it is dud, replace it.
If it is proven to be good, then the manual will tell you the colour marking of the ignition wire. Check it to see if it is OC [open circuit] If it is, then fixing the break, or re running a new one should solve the problem.