You have come to the right place with your question.
The black barrel is your coil and it is nowhere near the points.
Your points are located behind the flywheel and a bit of them rub on the cranksahft.
page 18 shows someone going after the points on a Garelli
I don't know gadabout, but its' gotta be similar.
to just look at, or check the gap, or even clean the points a bit, you can do it from one of the slots in the flywheel.
to change them you have to pull the flywheel off (pic 13 shows it gone) and that's not always easy. may take a special puller. Don't try to pry it off, things are too delicate.
I hope someone with a gadabout will add to this.
Hve you read Fred's guide under resources, articles at the top of this page?