I agree in general, though afaik there is just no really rigid theory for what constitutes a focal point—it can be anything that is salient. If you let people play in a lab and give them game matrices with multiple equilibria with identical payoffs, then coloring one equilibrium can make it focal point; but in reality many things can seem salient. Maybe it’s somehow built into our genetic and cultural code what we coordinate on—e.g. what’s best for “all” or what’s best for “our group” etc. (IIRC, Ken Binmore suggests something along the lines of “Evolution makes us find Nash bargaining solutions fair” in the book Natural Justice, but I don’t remember what his evidence is to support that.)
Concerning symmetry and Nash: you can model the Nash bargaining solution asymmetrically, but of course it’s unclear whether that helps. Models like Rubinstein’s are elegant but not really realistic in their assumptions and neither in their implications.