What is wrong with this $\mathsf{L} \subseteq \mathsf{L}-$uniform $\mathsf{NC}^1$ argument?
In fact, the circuit depends on the input structure, not only on the input structure size. We take a tree-decomposition of the graph with additional colours and turn it into a convolution tree. The evaluation of the formula on this tree is reduced to computing the value of the convolution tree. To compute the value of the tree, it is turned into an arithmetic circuit. Hence we do not get one circuit for each input size as required for $NC^1$, but rather one circuit for each single input.