"Not exactly" can have different implications depending on context. .
Let's go through these possible translations:
1) "nicht gerade" / "nicht genau"
I agree with your tutor, I would never use "Nicht gerade." as a short phrase on its own, that's not idiomatic. "Nicht gerade" has its place in a sentence when ...
It's possibly a generalization from the literal meaning any amount, indeed.
It works for example to say that the big man will pay any amount, which implies practically, for all intents and purposes, a hefty sum.
A more convoluted derivation requires a fossilized phrase with independent development of the individual components outside the phrase. In ...