[ColinWright]

The entire point of the post is exactly to show what the Banach-Tarski Theorem is adding to our understanding.

Summarising ...

Previously, people thought that it might be possible to define a measure on arbitrary sets and have the usual desirable properties of isometry invariance and finite additivity (we already know countable additivity won't work).

But the Banach Tarski Theorem shows us that's not possible. That is thereby adding to our understanding of what is and is not possible with measures.

So yes, it does add something to our understanding.