You shouldn't be measuring ACROSS the coil terminals.
The points being open or closed WILL affect the reading because you're essentially measuring coil resistance (which is very small).
If you measure from either coil terminal to ground with the points open, you'll get battery voltage at both. If the points are closed, you'll get a lower voltage reading on the terminal going to the points, which are basically grounding the circuit.
The question is, does it run?
If you did the test in your last sentence, then that means you're getting spark to the distributor cap, but you may not be getting spark to the plugs if the distributor cap has a bunch of crap built up inside it that provides an alternate path to ground other than to the plugs. Maybe the spring on top of the rotor that makes contact with the center of the cap is broken off.
