Yes, it's a natural / common expression.
The air traffic controller tried tocould not talk him out of it
means:
The air traffic controller tried to persuade him not to do it but he was not persuaded.
I don't see a suggestion of panic in the excerpt.
He might've responded to the air traffic controller; he might not -- the excerpt doesn't say.
Why passive?:
consider these alternatives:
"The pilot could not be talked out of it"
"The air traffic controller could not talk the pilot out of it"
The relevant / important thing is the pilot / the pilot's attitude, not the air traffic controller, so the pilot is the subject of the sentence, which makes makes the passive the natural choice.
"Can I replace “can’t be talked out of it” with “can’t be contacted with“ or “can't be called up”?"
No -- different meanings.
Ps. In the excerpt, he couldn't be talked out of it. Maybe obviously, somebody can be talked out of something -- they can be persuaded not to do something that they intend to do.
e.g: "I talked him out of it" / "I managed to talk him out of it"
And somebody can also be talked into something. I.e. -- persuaded to do that thing :)