But "he is moving" can be הוא זז/ נע as well
Think it would be useful to have a less ambiguous test...
Can we say הוא עובר בית?
Yes, but I think it's less common.
הוא נע. הוא לא מת
True, נע or זז are "moving". עובר is here in the sense of moving to a different flat or city.
Why not עובר ל דירה ?
I believe it is idiomatic the same way "move house" is in British English.
He's just pining for the fjords.