I think I read the requirement to separate ammunition from the firearms during transit in separate luggage is some sort of idea, that if some obtains and steals the gun case, they don't have the ammunition immediately available as well.

Whether that is a sensible reasoning, I didn't make the rules up.

USA may have different rules.

Most times when I check in, my duffel with ammo bix inside gets declared and I am asked hoiw it is packed. usually it goes through normal luggage lines. Maybe tagged somehow. The gun case after weighing etc, goes to a special channel. It might be opened an inspected to ensure it is unloaded. Actually not a bad idea, there are idiots in this world. Then off it goes. Locked. No one needs to access it until it is checked over at the destination.

For international travel, normally a visit to the Customs office before is required to have it checked over per the paperwork.

BTW I have a Tuffpak. I bought it years and years ago. But never used it for air travel. I think it would blow my kg allowance for checked luggage. Years ago and maybe still Americans had very generous allowances most of the world did not. Since then even the gun case for us is often or usually included in the 20 to 30 kg allowance (depending on the airline and flight). Tuffpaks are too damned heavy even empty. But I couldn't use just a Tuffpak with rifle and clothes, everything inside one. I use a duffel and a gun case. Always have other stuff to take.

I do use my Tuffpak on driving internal trips. It was supposed to fit nicely in the back seat of my Landcruiser and make it easy to slip firearms in and out for use. And them being locked away, out of sight and secure. Of course nothing goes to plan.

The lid of the Tuffpak is awkward to put on and off. Also the back doors fo the Landcruiser doesn't open enough to slide the firearms out. So the whole thing and anything on top of the case needs to lifted up to facilitate thje right angle. Putting in at the back, again the door of the Landcruiser doesn't line up at all. The cargo area in the back is not big enough for the huge Tuffpak, and I have cargo barriers dividing that area up anyway etc. In theory things always work so well. In practice, sh^t happens.