Why doesn't the model prompt engineer itself?

Because it is a challenging task, you would need to define a prompt (or a set of prompts) that can precisely generate chain-of-thought prompts for the various generic problems the model encounters.

And sometimes CoT may not be the best approach. Depending on the problem other prompt engineering techniques will perform better.

By this point, instruction tuning should include tuning the model to use chain of thought in the appropriate circumstances.