> The Algerian boxer was born female, but has high testosterone due to whatever medical condition,

Imane Khelif has an X and a Y chromosome. She has 5-alpha reductase deficiency, which leads to the development of a pseudo-vagina and internal testicles. Crucially, though, the hormone levels are the same as typical males. In terms of upper body strength, red blood cell count, bone density, etc. Khelif is the same as other males.

She wasn't disqualified due to hormone levels. She was disqualified because the International Boxing Association's criteria for participating in the women's category is having a female karyotype (no Y chromosome).

> Do you want sports governing bodies to inspect genitalia? Do blood tests?

Chromosomes can be checked with just a mouth swab.