Saying that the determinant is useless to determine whether a matrix is incorrect, or misleading at best.

This is purely a matter of numerical stability. Of course [[1e-10, 0], [0, 1e-10]] is nonsingular, and its determinant is 1e-20, which does not equal zero.

Yes, when it comes to floating point issues we might want to use something else, and that's a valid complaint when it comes to NumPy code, but from a theoretical perspective the determinant is an excellent tool to determine singularity.