Intuition behind a 0% central/equal-tailed confidence interval At a vague intuitional level, I feel like, if we generated a 0%, two-tailed, truly central (or "equal-tailed") confidence interval (CI), the number it would wrap around would be a median-unbiased estimate of the population parameter in question.  Am I confused, or is this right?  Is there another intuition of it that is better?
I can't seem to find anything about it when I google, but my intuition rests on the idea of the CI as the inversion of a p-value.  I think, this CI would also match one end of a one-sided 50% CI, which would also be the inversion of a 0.5 one-sided p-value.  And a 0.5 one sided p-value is saying "if this were our parameter, then 50% of datasets would be look to be 'higher' and 50% of them 'lower'".  So I feel like my intuition is right, but I can't quite formalise it, or find anything online to confirm or deny this.

You are close
For a continuous distribution, the 0% equal-tail CI occurs at the point corresponding to the median of the true distribution of the pivotal quantity that is used in constructing the CI.  It is not always possible to invert the pivotal quantity in a way that yields an unbiased estimator of a corresponding median (of what exactly?).
