Why can't we use the semi-minor axis in Kepler's third law?
The relationship between $a$, $b$ and the eccentricity of an orbit $e$ is
$$ a = b\left(\frac{1+e}{1-e}\right)\ .$$
It is clear therefore from a mathematical point of view that you cannot get a similar constant relationship by replacing $a^3$ by $b^3$ in Kepler's third law. You would have to use an expression involving both $b$ and $e$.
Another way of saying this is that you can have (infinitely) many orbits that have the same $a$ and $T$, but which have different $b, e$ combinations.
I guess from a fundamental point of view it boils down to being able to express the total energy of an orbit as
$$ E = -\frac{GMm}{2a}$$
which is independent of the shape of the ellipse.