When the engine is not running, the voltage regulator points will be closed. The cut-out points will be open. Confirm we are really looking at the voltage regulator points. When the engine is started and is run up to enough speed to charge the battery, the cut-out points will close. The voltage regulator points should remain closed until one or the other of these two things happen:
1) The current produced by the generator exceeds the limit.
2) The voltage produced by the generator exceeds the limit.
In the normal recharging of a moderately discharged battery, neither of these will happen and the generator will be producing (maybe) 15 amps and a voltage somewhat above whatever the battery was producing. As the battery charges, voltage from the generator will gradually increase until the battery is fully charged and event #2 happens.
When either of the two events happens, the coil under the voltage regulator points should produce enough pull on the points for them to open. When they open, the field circuit is open and the generator stops producing electricity. When that happens, the voltage and current from the generator goes away allowing the points to close. Then the generator starts charging again and the cycle repeats. This cycle should happen so rapidly that you don't see the cycling on the ammeter. The problem in your case seems to be that the cycling is so slow that you can easily see it. This suggests the voltage regulator coil/points assembly isn't sensitive enough to respond rapidly. Maybe the large gap you observed is part of the problem. If you could measure the voltage required to open the points, you may have a clue to the problem. It may be opening at too low a voltage, causing the cycling with a partially charged battery.
Note: I said points open cause the generator to stop. In practice, the regulator probably has a resistor in parallel with the points so the generator doesn't stop charging, but cuts back to a low charge rate.