I haven't worked on film sets, but I've worked on theatrical productions which utilized blank-firing guns. The ones we specifically used had been modified to prevent the possibility of live ammunition being loaded into them, but that's not always possible (like in the specific scene being shot for Rust, where the gun was to be pointed straight at the camera).

Generally, as others have said, the actor checking the gun is not part of the process because it adds additional risk, and may mess up the setup the armorer did. When we used blanks, the process was that the armorer would check and verify the gun was in safe and working order and was loaded correctly (i.e, for the productions we did, loaded with exactly two blanks, since the gun would be fired twice, and the revolver set to fire those two blanks in the correct timing). He would then place the gun in the specific spot on-stage where it would be retrieved from by the actor during the show (or directly hand it to the actor as they went on-stage for one of the shows). At that point, crew was not allowed near that spot on stage, and the only people allowed to touch the gun was either the armorer or the actor. Additionally, the armorer stood watch off-stage for the entire time the gun was out of his possession, and ensured nobody came near the gun except for the actor who was supposed to be using it as part of the show. After the gun was used in the show, he would immediately retrieve it, ensure it was rendered safe, and it would immediately be returned to the secure storage location we had for the gun. If we had ever run into an issue where crew would need to access the area that the gun was placed on-stage, the armorer would have removed and made safe the weapon before any of the rest of the crew could access the area.
Crew was made aware through explicit call-outs when each step occurred-- so when the gun was loaded, that was called out to crew via comms. When it was placed, that was called out. A call-out was made when the gun was retrieved by the actor, and again several seconds before it was to be used. And finally, a call-out was made when the armorer retrieved the gun and made it safe.

This is the same process that every stage or screen production is supposed to have. The gun is never, ever, ever to be used with live ammunition for any reason. Live ammunition should never even be on the set, for any reason. The gun should never be passed off to anyone else other than between the armorer and the actor (or on large enough productions, the armoring team under the direct supervision of an armorer). Nobody should be stopping to inspect the guns, because that means people who are not qualified will be handling the guns outside the control of the armorer.

Currently, firearm safety on set isn't everyone's job. But it should be everyone's job. The system should be better, because firearm safety is a demonstrably life-or-death process.

Yes, firearms safety is a life-or-death process, but that's precisely why the rest of the crew and actors don't need to have firearms be their job. All they need to know is that they do not touch any of the weapons, for any reason. If it's out of place, they should ensure nobody comes near it, and call the armorer to retrieve it. The chain of custody for the weapons must be incredibly short, and you don't want anyone who is not specifically authorized to be touching or interacting with the weapons in any way, because that's how mistakes start to happen.

The weapons should only be outside the direct control of the armorer for the minimum time possible, and the armorer should be observing the entire time. As soon as the scene finishes, and between shots, the armorer should take control of the weapon again and do all the steps required to ensure it is safe.

You don't hold the actor Alec Baldwin responsible. You hold the producer Alec Baldwin responsible.

Agreed, along with the rest of the producers. Concerns had been expressed about the armorer previously, and the production team should have responded to those and found another armorer who could safely manage the weapons (or, since some of the articles I've read suggest the armorer didn't feel she could push back on other crewmembers when they wanted to do things incorrectly for the sake of timing, they should have made it clear that when it comes to how weapons are used on set, the armorer is the voice of god and has the final say at all times).

Additionally, the production team should have found other ways to film the scene where the gun was looking down the barrel of the gun, by either using mirrors to ensure the camera and crew were not in the line of fire, or by filming it remotely. Since the cinematographer was shot during a rehearsal, a rubber replica should have been used to set the focus and framing for the shot, and the live weapon should have been swapped in at the latest possible moment before filming commenced.