Many modern guns for personal protection do not have manually actuated safeties and instead rely on other mechanisms to prevent accidental discharge. For example, glocks have a "trigger safety." Without getting into minutia, these sort of features do a fine job of ensuring the gun does not fire unless the trigger is intentionally actuated.

The reason manual safeties are going away is that for side arms, time matters A LOT. Gun fights are typically over within just a few seconds and the person who fires first has a huge advantage. A long time ago there were quite a few major incidents involving police and manual safeties which resulted in most departments changing their policies such that they carry with the safety off or, typically these days, glocks which have no manual safety.

Regarding the p320. For one, the 320 does not have a trigger safety. In fact, there are numerous "innovations" on this firearm which were made to cut costs, improve trigger feel, and various other things. This rose a lot of suspicion when the gun first came out actually - especially the lack of a trigger safety which many consider essential for striker fire guns intended for carrying on your person. On top of that, sig did end up making a p320 with a safety as this was a military requirement. The thing is, it is possible that the "innovative" sear mechanism within the p320 may allow the gun to fire even when the safety is engaged. In fact, there have been reports of just that happening.

The gun that was being carried in Air Force incident is the military version of the p320 which does in fact have a manual safety. Also, the leaked reports state the incident happened whenever the gun was in the holster and the holster was removed and placed on a desk facing the victim. This is when the gun misfired striking the victim in the chest. There is speculation that the gun may have in fact had the safety engaged whenever this happened. If so, this would explain the prompt response by the AirForce.

Regardless, the purpose of manual safeties are not to prevent guns from discharging on their own (aka. "uncommanded"). Manual safeties are intended to protect against "accidental" discharges where the trigger gets pulled "accidentally." That's not what happened in the above video or the AirForce incident.

One more thing to note. The officer in this video was using a Sig Sauer holster so there should be no risk of the holster itself being the problem.