There is a video available on the Israeli method of rapid slide manipulation, since they are required to carry with a dry chamber. Safe carry and it can be fast.
In the 1905 Colt catalog they show their new .45 pistol with the hammer resting on half cock, but there is no mention of this anywhere. Obviously it is perfectly safe to lower the hammer to half cock. However, the pistol was designed to be absolutely safe when the hammer is down on a loaded round.
It is odd that nobody has been able to reproduce an accidental discharge by dropping the pistol hammer down on a live round. Loading with a primed case only, a mechanical engineer with a PHD and I tried dropping the pistol from different heights. Odd fact: Only ONCE in 1,000 drops did the pistol land directly on the muzzle. Mostly it lands on its side. The pistol would not fire unless it fell from a seventh story window and landed directly on the muzzle, with the firing pin spring made by Wolff. Now, I'd say that was safe. But, weaker springs will fire with less inertia.
Browning's last design, the Radom pistol was designed without a safety, to be carried by cavalry hammer down on a live round, since they had to hold the reigns with their other hand. They cocked the Commander style hammer by stroking it across their thigh.
If carried on half cock, it is fatally easy to snag the hammer and have a cocked pistol safety off in your holster. It is less likely to snag to full cock from a fully down position.
One of the important things an instructor does is evaluate his pupils. It is your duty to keep individuals within their parameters of competence. I'm really glad to see that you are doing this.
Most ambi safeties will loosen up in time and can cause problems. The one that I like best is King's, since it has a dovetail to keep it from loosening.
One field test is worth a million speculations. Load a primed case and try to make the pistol fire by dropping it.
In all of the cases I have investigated, the "victim" of a drop fire accident was concealing some basic facts about the situation. But, don't take my word for it, actually try the test with 1,000 drops and report the findings to this forum.