Learn English – Similar idioms to “When the cat’s away, the mice will play”

The Time quote is sloppy. William McChesney Martin, Chairman of the Federal Reserve in the 1950s and 1960s, is quoted— in Time— as saying "I’m the fellow who takes away the punch bowl just when the party is getting good." (according to http://quoteinvestigator.com/2010/10/21/take-punch-bowl/ )

Punch, at least the punch one would serve in the U.S. at parties for adults, is often made with bourbon or rum. So according to Martin, the role of the central bank is to restrict the money supply to slow growth just as the economy starts booming (to everyone's annoyance), just as one might take away the alcohol just as a party begins to rock (to everyone's annoyance).

It is not a "general purpose idiom," but the reference is well-understood in financial or economic circles because another chairman, Alan Greenspan, used it in a speech more recently.

The expressions need to be considered in context here:

The first example "Engage the clutch before selecting a gear." refers to pushing the clutch pedal inwards from its normal rest position, thereby engaging it.

Releasing the clutch to move the vehicle forward has to be a smooth operation (not necessarily slow) to avoid stalling the vehicle.

The second example "Laura let out the clutch and pulled slowly away down the drive." and "I have to learn how to let the clutch out smoothly." refer to releasing the clutch from its engaged position to move the vehicle forward.

So the right way to look at these expressions is from the perspective of the user (the driver) and not the mechanical operation of the clutch.