Let $G$ be a semisimple algebraic group and $\Sigma$ a smooth proper curve. Then $\text{Bun}_G(\Sigma)$ comes equipped with a line bundle $\mathcal{L}$ which generates the torsion free part of $\text{Pic}\text{Bun}_G(\Sigma)$ (e.g. in type A or C it's the determinant bundle or in type B or D the Pfaffian bundle). Then the Verlinde formula is an explicit formula for $$H^0(\text{Bun}_G(\Sigma),\mathcal{L}^{\otimes k}),$$ intimately related to fusion products.

There are lots of nice proofs of the Verlinde formula published in the 1990s, e.g. Beauville's Conformal blocks, fusion rules and the Verlinde formula. However, they are all quite algebraic which makes it hard (at least for me) to understand what's going on. Given how much better the geometric side has been understood in recent decades (e.g. fusion and the BD Grassmannian), is there written up anywhere a slightly cleaner/more geometric proof of the Verlinde formula?

Let $G$ be a semisimple algebraic group and $\Sigma$ a smooth proper curve. Then $\text{Bun}_G(\Sigma)$ comes equipped with a line bundle $\mathcal{L}$ which generates the torsion free part of $\text{Pic}\text{Bun}_G(\Sigma)$ (e.g. in type A or C it's the determinant bundle or in type B or D the Pfaffian bundle). Then the Verlinde formula is an explicit formula for $$H^0(\text{Bun}_G(\Sigma),\mathcal{L}^{\otimes k}),$$ intimately related to fusion products.

There are lots of nice proofs of the Verlinde formula published in the 1990s, e.g. Beauville's Conformal blocks, fusion rules and the Verlinde formula. However, they are all quite algebraic which makes it hard (at least for me) to understand what's going on. Given how much better the geometric side has been understood in recent decades (e.g. fusion and the BD Grassmannian), is there written up anywhere a slightly cleaner/more geometric proof of the Verlinde formula?

Edit: I'm mainly curious about whether there is a "more geometric" version of Beaville's proof, but am very happy to see other methods also.