Bhargava and Shankar have proved results about average $3$-Selmer ranks of elliptic curves. (See this arxiv preprint, which is
the same paper cited by the Wikipedia article linked in the OP.) Their
argument is via geometry of numbers (so to speak).
In fact, they are able to construct families such that exactly half of them
have positive sign in their functional equation, and with average $3$-Selmer rank bounded by $7/6$. (This
can be achieved by imposing appropriate conditions on the coefficients of the
elliptic curve, and computing the root number as a product of local roots numbers.) Now
work of the Dokchitser brothers on the parity conjecture implies that for elliptic curves with sign $+1$, the rank of the $3$-Selmer group is even. When combined with the
bound of $7/6$, they deduce that the $3$-Selmer groups of the curves with sign $+1$ that lie in their family must be trivial.
Now (under some additional assumptions about the $3$-torsion, and some other
technical assumptions, which they are able to impose on their family) by applying the results of Skinner and Urban on the Main Conjecture (which lets one pass from triviality of a Selmer group to non-vanishing of the $L$-function) they deduce that the curves in their family having sign $+1$ also have non-vanishing $L$-value at $s = 1$.
Now a positive proportion of elliptic curves overall lie in their family,
and so putting all this together, one finds that a positive proportion of elliptic curves have both $3$-Selmer rank zero (and in particular, Mordell--Weil rank zero) and also analytic rank zero. Thus, a positive
proportion of elliptic curves satisfy (the rank part of) BSD.
From my brief reading of the paper, the work of Kolyvagin and Gross--Zagier is not actually used (at least explicitly); Wikipedia seems to be in error on this point.