Not knowing anything about guns I would assume you would engage the safety when putting the weapon in the holster. Is that not true?

~"It's complicated".

There's really 2 different things here:

Firstly, there's "How many things are between my finger on the trigger and shooting the gun". The furthest you get is not having a round in the chamber, where you have to pull back and release the slide to chamber a round. In that state, basically nothing you do to the trigger or any other part of the gun will result in a discharge. You could use the gun to hammer a nail into a board and it would be really inefficient but also not result in you shooting anybody. From there, you can have a round in the chamber but the safety on. Some guns have no safety. Some guns have more than one (a common combination is a toggle safety and then a bar built into the trigger that must be pulled first before the trigger can be pulled). The safety's job is to stop the gun from slamming the firing pin into the back of the round. All of this matters a lot for the kind of issues that were common with Serpa holsters, where users tended to slide their finger along near the trigger and were accidentally pulling it as part of their draw from the holster.

Secondly, there's "is there anything stopping the firing pin from just smacking the round and firing it w/o the trigger or anything else being involved". In some guns, the firing pin is physically blocked from striking the round until the trigger is pulled back: there's a piece of metal or other impediment that's in the way, and when you pull the trigger it slides out of the way and then the mechanism pushes the firing pin forward. But in other guns, that isn't the case: the firing pin is held away from the round by some style of tension, but isn't physically blocked. On those guns, if you have a round in the chamber and you whack the gun in the wrong direction, the firing pin can push into the round and fire it. Sig's prior claims were that this was not possible on the P320. Evidence suggests that they are incorrect.

This is a hard question to answer, but I'll try to boil it down to this specific firearm. Many handguns, like all Glocks and many Sigs, these days are "striker fired," meaning there's an internal hammer that strikes the hammer versus an external hammer. Internal hammers are more complicated than external hammers, and that includes, in the case of Glocks, two built-in safeties internally that would prevent misfires. The only way they would fire is if the trigger is pulled. Period.

There are no external safeties for the operator to engage with these handguns. They will only go off the trigger is pulled - so drops should not set it off, nor the actions in the video. You have to intentionally pull the trigger for the gun to go off, which is the ultimate last word in safeties.

There are still semiautos with external safeties hammers, the most famous being the 1911. These are what's called single actions. The trigger weight (amount of pull on the trigger) is relatively light, so they have an external safety for the operator to engage/disengage.

I personally prefer single actions, hammer cocked, safety engaged, but this is always a very, very personal preference by people that carry. I own Glocks, but I would not carry one because of the lack of external safety, however, I would never criticize anyone that does. This is 100% strictly boils down to what the person is comfortable doing.

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.

The short answer is that modern handgun design leans away from safeties that must be actively toggled by the user in favor of automatic safeties and interlocks that are transparent to the user, like the seat switch in a riding mower. Manual safeties still exist but they're considered an accessory feature.