How much would Bhargava's results on BSD improve if finiteness of the Tate-Shafarevich group, or at least its $\ell$-primary torsion for every $\ell$, was known? Would they improve to the point of showing $100$% of elliptic curves over $\mathbf{Q}$ satisfy the BSD conjecture? (which, of course, would still not prove the BSD conjecture)

I've just attended a very nice seminar talk about the topic, and I'm curious to get some expert info.