Why does it make sense to use min-entropy for measuring cryptographic random data?
Ignore NIST. It has more important legally binding agendas to debate outside of this forum. Use common sense.

Does min-entropy satisfy the properties of an entropy measuring function laid out by Shannon? If not, why is it being used for measuring entropy?

No. $H_{\infty}$ is to Shannon entropy as is JPEG to BMP. Shannon was interested in encoding all the data in the transmission channel.  $H_{\infty}$ is lossy and the original data cannot be retrieved in it's entirety. $H_{\infty}$ is used before transmission.

entropy is produced in raw random data

Yes. So measure it. You will have a lot more luck if you can prove the data to be IID.
It boils down to this:-

You need to separate information entropy from cryptographic entropy.
We want true random bits.
We get them via some generic form of hardware processing.
Current mathematical thinking revolves around the left over hash lemma which dictates the eventual output bias.
The thinking is predicated on $H_{\infty}$.
Thus $H_{\infty}$.