The problem with the determinant is not about performance. It is just useless for determining if a matrix is singular. The thing that gives it away is that the determinant is influenced by a rescaling of the matrix:
det(s A) = s^n det(A) where A is a n x n matrix
As an example, would you say that [[1e-10, 0], [0, 1e-10]] is singular? It has condition number 1.
Let's say we are in a setting where we only work with integers. A matrix is invertible iff its determinant is invertible in the underlying ring. The only invertible elements in Z are -1 and 1.
So, the code is also incorrect in the integer setting. Here, we should not check for 0, but for -1 or 1.
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.
Numerically, such a binary answer is pretty useless. Here, we need a measure of how singular/nonsingular a matrix is relative to the numerical precision we are working with.