There are many type of sections. First the 9 exceptional divisors are sections. Then the lines through 2 of the 9 base points are sections, conics through 5 of the 9 base points are sections, cubics through 7 of the 9 base points, which have a node at one of the 7 points are also sections, etc.

The Picard group of $S$ has rank 10, hence there are at most 9 disjoint -1 curves. If $C_1$ and $C_2$ intersect in 9 distinct points then there are 9 disjoint -1 curves. For rational elliptic surfaces the group of sections is finitely generated.

There is a minor issue with your argument: If $X$ is a smooth projective surface admitting a genus 2 fibration, with a singular fiber with more than 2 irreducible components then $X$ is only birational (as $\mathbb{P}^1$-variety) to a double cover of a ruled surface. Even if you read the OP as X admits a fibration in genus two curves and there are precisely 7 singular fibers, each of them has one node (and therefore at most two irreducible components) your argument does not always work.