Hi Adam

I'll think about this a bit and get back. I haven't touched one of these for over 20 years so it's a fuzzy memory.
A couple of thoughts to start...

1. The solenoid moves a brake arm. the arm does 2 separate jobs - engage with the drum brake, and activate the clutch. The arm will have a finger on it (maybe like a plastic pointer) that pokes into a set of plastic teeth, like the teeth on a gear wheel, that surrounds the clutch.When the arm snaps back at the end of spin, it grabs the brake to stop the drum ,but another part of the lever protrudes between a couple of the plastic "gear teeth" and rotates the plastic slightly. This is what releases the clutch.
2. The clutch is a simple spring. The last maybe 10mm of the spring sticks out and engages with a slot in the inside of the plastic "gear wheel".

I wonder if this 10 mm sticking out bit of the spring has snapped off?

Or maybe the "poking finger" of the brake arm is broken?