The reason that your interesting post reminds me of my (voted to be closed!) post I mentioned above is that the philosophy of Hahn Banach is to extend certain functional to a bigger space. here you are searching for an appropriate bigger space. (My +1 to your interesting post)

This remind me of a question of mine I asked at MO about 8 years ago the content of my post (which was voted to close!) was the following: "We consider the classical space BV[0,1] of bounded variation functions the total variation is a subadditive function. Are there some application of Hahn Banach theorem to this particular subadditive function?

@RobertBryant Is it a good idea to introduce a generqlized "configuration space" as follows: All n tuples for which the corresponding vector function(squared) is a topological embedding (or in case of empty cut locus case one may consider all n tuple for which the corresponding squared map is isometric or symplectical embedding). In case of negative curvature does it generate a configuration space worth of study?

@RobertBryant Thank you for introducing me the cut locus point. I was not aware of it. Are they same as focal points (or any realtion to it). I think in negative curvature we have more chance for smoothness of the squared distance function yes?

Any way I vaguely remeber the square distance function, distance to a set not to single point, is introduced as a real analytic function(in a book by S. Krantz whose title contains (I thinl) "...real analytic....". I do not remenber the reason this function was introduced in his book but may be his consideration would be helpful in line of your question

+1 for you question I think it generates more questions on some particular subcategory of smooth manifolds: For examples what is a precise example of a symplectic manifold and a finite even number of points for which the square distance vector function f you mentioned is a symplectic embedding?

@RobertBryant let's consider the square of distance instead of disrance as you pointed out to. I wonder can one prove the easy or advanced Withney embedding theorem or Nash isometric embedding theorem via such function $f$? More precisely: Assume that $(M,g)$ is a Riemannian manifold are there a finite number of pivot points for which the function $f$ (with square distance) would be an isometric embedding? I did not read the Nash proof but I wonder is his proof based on such a function?