This is addressing the second part of OP's question: could the critical locus be smooth for $n>2$? Given an $(n+1)$-tuple of degree $d$ polynomials, $(F_0,\dots,F_n)$, in $n+1$ variables, $x_0,\dots,x_n$, the Jacobian matrix $[\partial F_i/\partial x_j]$ defines local morphisms from $\mathbb{P}^n$ to the affine space of $(n+1)\times (n+1)$ - matrices. The critical locus of the morphism is the inverse image of the determinant hypersurface under this morphism.

Since the determinant hypersurface is singular in codimension $3$, i.e., codimension $4$ in the ambient space of matrices, the inverse image of the determinant hypersurface "should" be singular in codimension $3$, i.e., codimension $4$ in $\mathbb{P}^n$. In fact, using the Thom-Porteous formula, the degree of the codimension $4$ part of the singular locus equals $$\frac{(n+2)(n+1)^2n(d-1)^4}{12}.$$

This is addressing the second part of OP's question: could the critical locus be smooth for $n>2$? Given an $(n+1)$-tuple of degree $d$ polynomials, $(F_0,\dots,F_n)$, in $n+1$ variables, $x_0,\dots,x_n$, the Jacobian matrix $[\partial F_i/\partial x_j]$ defines local morphisms from $\mathbb{P}^n$ to the affine space of $(n+1)\times (n+1)$ - matrices. The critical locus of the morphism is the inverse image of the determinant hypersurface under this morphism.

Since the determinant hypersurface is singular in codimension $3$, i.e., codimension $4$ in the ambient space of matrices, the inverse image of the determinant hypersurface "should" be singular in codimension $3$, i.e., codimension $4$ in $\mathbb{P}^n$. In fact, using the Thom-Porteous formula, the inverse image of this codimension $4$ locus has degree $$\frac{(n+2)(n+1)^2n(d-1)^4}{12},$$ at least assuming that the inverse image has codimension $4$ (i.e., not codimension $0$, $1$, $2$ or $3$).

This is addressing the second part of OP's question: could the critical locus be smooth for $n>2$? Given an $(n+1)$-tuple of degree $d$ polynomials, $(F_0,\dots,F_n)$, in $n+1$ variables, $X_0,\dots,X_n$, the Jacobian matrix $[\partial F_i/\partial x_j]$ defines local morphisms from $\mathbb{P}^n$ to the affine space of $(n+1)\times(n+1)$-matrices. The critical locus of the morphism is the inverse image of the determinant hypersurface under this morphism.

Since the determinant hypersurface is singular in codimension $3$, i.e., codimension $4$ in the ambient space of matrices, the inverse image of the determinant hypersurface "should" be singular in codimension $3$, i.e., codimension $4$ in $\mathbb{P}^n$. In fact, using the Thom-Porteous formula, the degree of the codimension $4$ part of the singular locus equals $$\frac{(n+2)(n+1)^2n(d-1)^4}{12}.$$

This is addressing the second part of OP's question: could the critical locus be smooth for $n>2$? Given an $(n+1)$-tuple of degree $d$ polynomials, $(F_0,\dots,F_n)$, in $n+1$ variables, $x_0,\dots,x_n$, the Jacobian matrix $[\partial F_i/\partial x_j]$ defines local morphisms from $\mathbb{P}^n$ to the affine space of $(n+1)\times (n+1)$ - matrices. The critical locus of the morphism is the inverse image of the determinant hypersurface under this morphism.

Since the determinant hypersurface is singular in codimension $3$, i.e., codimension $4$ in the ambient space of matrices, the inverse image of the determinant hypersurface "should" be singular in codimension $3$, i.e., codimension $4$ in $\mathbb{P}^n$. In fact, using the Thom-Porteous formula, the degree of the codimension $4$ part of the singular locus equals $$\frac{(n+2)(n+1)^2n(d-1)^4}{12}.$$