Sharpen the question first: BSD asserts ord_{s=1} L(E,s) equals the rank of E(Q), and that the leading Taylor coefficient equals a specific product involving the regulator, real period, Tamagawa numbers, and #Sha(E). This is really two conjectures welded together — a rank equality and a refined formula — and they fail for genuinely different reasons.
What's actually proven is narrower than people assume. Gross–Zagier related the first derivative of L(E,s) at s=1 to the height of a Heegner point, and Kolyvagin used Heegner points to bound Sha and show that if the analytic rank is 0 or 1, the algebraic rank matches it exactly. This machinery is essentially the only known route from analytic to algebraic rank, and it says nothing at all about rank 2 or higher: there the L-function's vanishing order is not currently tied to any known algebraic invariant by an unconditional argument. Bhargava and Shankar's work on Selmer groups shows a positive proportion of curves have rank 0 or 1, so Kolyvagin's theorem covers a nontrivial slice of all elliptic curves over Q, but this is density, not the theorem.
Where does the real obstruction live? Finiteness of Sha itself is unknown outside the rank 0/1 cases handled via Kolyvagin's Euler system bounding Selmer groups. Without that finiteness, the refined BSD formula is not even well-posed. My position: progress on higher rank requires a genuinely new Euler system or an Iwasawa-theoretic input beyond Heegner points, not a strengthening of existing analytic techniques — the vanishing-order barrier at rank 2 looks structural, not merely technical.