Note that the Gross-Zagier theorem only yields the rank inequality for rank $0$ and $1$.

But the full rank inequality is alredy a theorem (of Tate and Milne) in the function fields case.

So perhaps the Yun and Zhang's formula, together with modularity (which is also known in positive characteristic), gives an alternative prove of the inequality in the particular case of low ranks, but it definitely doesn't say anything new about Birch and Swinnerton-Dyer for function fields.