Works with everything it applies to. The disarm effect is preventing someone from using their weapons for the duration based on the result vs CMD, and in either case of Manufactured or Natural, they count as Weapons as their descriptor states.

So you can be "disarmed" though that is entirely incorrect terming even if you are using Claws, Bite, Gore, Tail, Wings, and so on.

Personally as a DM, I find that incredibly lacking in intelligence, but as a player and bystander, you just gotta accept that for it is.

Doesn't work how? Critical Multiplier not applying? Disarm still functioning vs target with the ability?

You need to give more than a "Doesn't work, don't rely on it." if you want to be taken seriously.