Disclaimer : I'm not a native speaker, not even an advanced learner, this is just
my current comprehension of the わけ-expressions.
訳 can mean two very different things. When it is read やく, it's translation/interpretation, but when it's read わけ, it actually really means reason (as well as meaning/cause/result, it's kind of connected). And it makes perfect sense, I don't understand why you think it doesn't.
reason : a statement offered in explanation or justification, a rational ground or motive, the thing that makes some fact intelligible, a cause.
This one is complicated, and has been thoroughly analyzed here :
How to end a sentence in わけ
But just looking at the keywords in the graph there is "reason, cause, know fact, result". It all stem from わけだ meaning "it's the reason/cause for..." and has been extended in all kind of direction.
there is no reason for ... (No rational grounds, no logic, out of common sense)
It's a strong negation of a reason for someone to do something or for something to exist or be in some state. If a friend keep insisting that Santa Claus is real, at some point you can get annoyed and say :
There is no reason for Santa Claus to exist ! Santa Claus can't possibly exist ! There can be no Santa Claus!
The reason for ... won't go. (The reason for ... is not acceptable)
One cannot do something due to external pressure, usually common sense/society rules. Often translated "I can't afford to ..." to stress that we can't do something because it's not reasonable.
隣の部屋で今、赤ちゃんが寝ているので、ピアノを弾くわけにはいかない。 I can't afford to play the piano
because the baby is sleeping in the next room.
So here clearly it's not because I lack the competence that I can't play the piano, it's because it's common sense to not wake up a sleeping baby, so "the reason for playing the piano won't go" (is not acceptable).
Mildly deny the previous (and sometimes the following) statement.
Literally previous statement is not a reason for...
A: The weather forecast calls for clear skies tomorrow.
Literally B: (this statement is) not a reason for "rain won't fall"
More natural B: It doesn't mean it won't rain.
This was a quick tour and I'm sure I simplified and misrepresented a lot of things. The わけ-expressions are now idiomatic expressions, so they have been extended in all kind of way and the root meaning can be sometimes a bit blurry. Still I think all of them can be reasonably (^^) traced back to 訳 meaning reason.
Edit : By the way, I would like to add that this post is heavily based on the Dictionary of Basic Japanese Grammar, as well as this amazing blog post (a must read IMHO) https://nihongodaybyday.blogspot.com/2014/05/blog-post.html