I've taught around a dozen folks how to drive stick, and I finally found a method that reliably worked to make it "click" more quickly. When they get in the car, I tell them to forget about the gas and brake pedals and pretend the only pedal that exists is the clutch.

In first gear, I have them release it slowly until the car starts going, then have them push it in until the car rolls to a stop. Take breaks, explain what the clutch is doing. Repeat until they start getting bored.

Then introduce the brake pedal. Tada! You can stop more quickly now.

Finally, the gas pedal. You can push it until the engine shrieks in protest! If only there were a way to get it to quiet down but still keep your speed....

Of course, you need a very large, very flat surface for this method to work. But I've had a lot more success since starting to do it this way.