Learn English – “It isn’t” vs. “it’s not”

You might think there's a good reason for this, but there isn't. You said:

"There's not" sounds indescribably uncomfortable for most situations.

Are you sure that most doesn't mean, "for most situations I can think of off the top of my head right now"? I don't think most is a good word, but I might accept many – particularly if you mean, "for many situations where I use it in my first drafts."

As to "elucidating why this might be," perhaps it's because "there's not" might sound much improved if expanded to "there's no X", or if the sentence was restructured altogether:

You might think there's a good reason for this, but there's not.
You might think there's a good reason for this, but there's no good reason.
There's no good reason for this.

There's not any baseball player who went hitless for more than 40 games.
There isn't any baseball player who went hitless for more than 40 games.
No baseball player has gone hitless for more than 40 games.

There's not anyone there.
There isn't anyone there.
No one's there.

But that doesn't mean there aren't plenty of situations where the phrase works just fine:

"Are you crazy? We can barely meet our payrolls as it is! There's not enough work for the men we've got now!" (Ayn Rand, Atlas Shrugged)

"For the moment, though, suppose there's not just one, but two experiences here, each tactilely experiencing quite differently from the other. (Peter Unger, Oxford Studies in Metaphysics v. 1)

I don't think "there isn't" would improve either of those.

The issue is not whether "there's not" sounds more "wonky" than "there isn't", particularly for a single case. Remember:

"There isn't any single essence that binds all uses of language together." (Bryan McGee)

"Therefore, plywood is quasi-wooden".

"Therefore, plywood is pseudo-wooden".

"Therefore, plywood is semi-wooden".

"Therefore, plywood is only nominally wooden".