You're right that opening and closing the points should achieve the same thing as grounding the spring side of the points. Since that gives you a spark the condenser is not preventing the spark and probably it's OK.
The spark actually happens when the circuit opens, not when it closes, so you would have been seeing the spark from the coil when, after you have grounded the wire/s, you then lost the ground again.
The points could be either failing to make the circuit or failing to break it, but since you got a spark off the spring side the points must have been open, so we can assume they are failing to make the circuit, not failing to break it.
It's hard to see how the static contact would not be grounded, so it can only be either the contact tips are contaminated or they are adjusted too wide and not physically closing.
