You may think its getting 12V on the "run" position,but I bet someone used the original point ignition coil wire to power the HEI coil--that wire is a resistance wire and will deliver only 9 volts or so at reduced amperage ,which kept the points in the original distributor form burning up rapidly..there was a yellow wire from the starter solenoid that gave 12V while cranking to the original points coil to ease starting,that only got energized while the starter was cranking..
An HEI needs full 12V to work right...so you'll need to run a new wire from the fuse box from a source that is only hot when the key is both in the "run" and "crank" position..but not when its on "accessory",so the HEI wont get power when your parked listening to the radio,etc..you can determine that with a test lamp..
A quick test to see if thats your problem is to hook the HEI "Bat + " terminal directly to the battery positive terminal and see if it will start up and stay running..if it does,you need to run another wire to it as described..