For a compact manifold embedded in Euclidean space, the squared-distance function from a generic center is Morse

If $\dim M=0$, compactness makes $M$ finite and every function on it is Morse. Assume $m=\dim M>0$, so $N\ge 1$. Compactness supplies finitely many coordinate patches $x_j:U_j\to M$ on which there are smooth orthonormal normal frames $e_{j,a}(u)$, $1\le a\le N-m$. Each critical pair $(x_j(u),p)$ satisfies the equations in [A1], hence has a unique representation $p={\mathrm{\Psi }}_j(u,t):=x_j(u)+{\sum }_{a=1}^{N-m}{t}_a{e}_{j,a}(u)$. Thus the critical-pair manifold over this patch is parametrized by the open Euclidean set $U_j\times {\mathbb{R}}^{N-m}$, and the projection to centers is the smooth map ${\mathrm{\Psi }}_j$ to ${\mathbb{R}}^N$.

Put $H_i(u,p)=(p-x_j(u))\cdot {\mathrm{\partial }}_i{x}_j(u)$. At a critical pair, the derivative of $H=(H_i)$ in the $p$ direction has rank $m$, because the tangent vectors ${\mathrm{\partial }}_i{x}_j$ are independent. Hence a tangent vector to the critical-pair manifold lies in the kernel of the center projection precisely when $D_{u}H(u,p)$ has a nonzero kernel. But the coordinate gradient of $d_p$ is $-2H$, so its Hessian at that critical point is $-2D_{u}H(u,p)$. Therefore $p$ is a bad center exactly when it is a critical value of at least one of the finitely many maps ${\mathrm{\Psi }}_j$.

By [L1], each $\mathrm{CritVal}({\mathrm{\Psi }}_j)$ is null in ${\mathbb{R}}^N$. Their finite union is null and contains every bad center by step 2.1. Outside it every critical point of $d_p$ has nondegenerate Hessian, so $d_p$ is Morse. This finite-chart argument does not invoke the countable-choice-qualified manifold parametric theorem.