Firstly, the functional field result your state is due to Tate in his Bourbaki talk. In fact he proves that the finiteness of the $p$-primary part of Sha is enough for $p$ different from the characteristic.

For elliptic curves over a number field, the finiteness of Sha (over larger fields) gives us the parity on the BSD conjecture, by work of Dokchitsers.

The analogue of Tate's method carried over to the number field case is what Iwasawa theory for elliptic curves attempts to do. The corresponding result is considerably weaker. Here an example of what one can prove:

Let $E/\mathbb{Q}$. If we can find a prime $p$ of good ordinary reduction (just to make the statement cleaner) such that

- The $p$-primary of Sha of $E/\mathbb{Q}$ is finite,
- The canonical $p$-adic height is non-degenerate on $E(\mathbb{Q})$, and
- The orders of vanishing of the $p$-adic $L$-function and the complex $L$-function agree.

Then the order of vanishing of the complex $L$-function is equal to the rank of $E(\mathbb{Q})$.

As for the precise formula of the leading term, one only gets the formula for the $p$-adic $L$-function, subject to the first two conditions.