Why should you not apply the clutch when driving on a long, steep downhill slope?

On a long, steep downhill you should stay in gear and not press the clutch, because the engine then helps slow the car down. If you press the clutch, the connection between engine and wheels is interrupted, so the engine braking effect disappears and the car rolls downhill more freely. That means you must control your speed almost only with the foot brake.

On long descents this is a problem because continuous braking creates a lot of heat. The brakes can fade (their effect becomes weaker), the braking distance increases, and in extreme cases you can even lose braking power. Using engine braking reduces how much you need the brakes, keeps them cooler, and gives you safer, steadier speed control.

The statement about the brake servo no longer functioning is not generally correct for modern cars in normal driving: the brake booster is not “switched off” just because you press the clutch. The key rule of thumb for similar questions is: downhill, choose a low gear early and let the engine brake; use the foot brake only to adjust speed, not to hold it continuously.