But do you understand the problem and its context well enough to write tests for the solution?

Take prolog and logic programming. It's all about describing the problem and its context and let the solver find the solution. Try writing your specs in pseudo-prolog code and you will be surprised with all the missing information you're leaving up to chance.

I am not writing the tests, LLMs are.

My objective is to write prompts for LLMs that can write prompts for LLMs that can write code.

When there is a problem downstream the descendant hierarchy, it is a failure of parent LLM's prompts, so I correct it at the highest level and allow it to trickle down.

This eventually resolves into a stable configuration with domain expertise towards whatever function I require, in whatever language is best suited for the task.