B-F, You said this was your first EM, so I'll be very basic.
There are actually 3 switches involved with a single slingshot. There are 2 switches that protrude vertically on the top of the playfield and sit right behind the slingshot rubbers. When they close, these switches deliver DC power to the solenoid that operates the kicker arm. The 3rd switch is mounted right underneath the playfield near the top of the plunger for the kicker arm? That is the switch that actually routes pulses to the the 10-pt relay. It is normally open, but closes when the plunger for the kicker arm gets pulled into the solenoid. See the photo below.
That switch is usually the culprit when a sling kicks, but does not score points. Inspect that switch, clean the contact faces, and make sure they close when you manually depress the plunger into the solenoid core. Also check that all wires attached to the tab ends. See if that fixes the issue.