When we talk about integrability of classical systems in terms of Hamiltonian mechanics, it's all to do with counting independent conserved quantities.
Then when we move to the Hamilton-Jacobi formalism, suddenly everything is about separability of the Hamilton-Jacobi equation and Staeckel conditions. How do these two concepts relate to one-another? Does the existence of a certain number of conserved quantities imply separability of the Hamilton-Jacobi equation in some coordinate system?
The answer to your question is yes, the existence of $n$ conserved quantities with $n$ degrees of freedom implies separability of HJ.
The massless HJ equation is $$g^{MN}\frac{\partial S}{\partial x^M}\frac{\partial S}{\partial x^N}=E.$$ It separates if there exists a new set of coordinates $Y^M$ such that $$ S(Y_1,...,Y_n)=\sum_{i=1}^n S_i(Y_i),$$ which implies existence of $n$ conserved quantities, because each term in the HJ equation depends on its own variable. The same procedure is used when we solve PDE. For example in $2D$ $$S=S_x(x)+S_y(y), \quad f(x)(\partial_x S_x(x))^2+f(y)(\partial_y S_y(y))^2=E.$$ The latter means that both terms in LHS are constants separately. So we have two independent conserved quantities.
