The thing is how would they have even constructed the phrases in the first place?

To use the word 'and' to construct the phrase in the first place, it already had to mean combining the two. Similarly "at" has to have the meaning to specificy the location. On top of Darmok, Jalad, and Tanagra all having names independent of the phrase.

We couldn't have done "Juliet on her balcony" without being able to understand what a balcony even is or Juliet independent of the scene. This also means we can take the larger narrative and break it down into that key piece for the metaphor.

Maybe instead they once had a language and it eroded away into just memes, but then how did they manage to synthesize "Picard and Dathon at El-Adrel" in the end?

I concur. I am not a young person. And yet, I have an idea of what "rule" means in the title of a shitpost, and I shouldn't even know what "shitpost" means. I have never seen a formal definition for either of them. I don't think "rule" (in that context) even has a rigid definition yet. The meaning isn't just communicated by example; it is being invented by example.

I still feel like there has to be a really big turtle at the bottom who is not standing on another turtle. Words that mean things in themselves. But I've never been there.

The whole point was that they communicated through metaphor, which they aren't doing if they simply memorize the sounds. That would make "Darmok and Jalad at Tanagra" the literal words that mean "working together" like words in any language. To speak in metaphors they don't have to know the whole story, but they at least have to know it's a reference to two people working together.

A C++ user could never make novel assembly that C++ couldn't already target without also being able to deal in assembly.

So your concept may explain all the speaking in only old metaphors, but in the end they synthesize a new one: Picard and Dathon at El-Adrel. So they clearly could still use the underlying languag.

Also, Dathon listened to Picard tell a story, and Dathon seemed engaged with language as a narrative. This means Dathon was able to grasp the story as constructed from words he ostensibly wouldn't have understood?

We can learn C++ without without knowing binary because we have C++ documentation written in human languages. Likewise, the Tamarians need a non-metaphorical language to explain their metaphors.