What are bounded-treewidth circuits good for?
We now understand that for any fixed bound $k \in \mathbb{N}$ on the treewidth, we can convert any Boolean circuit of treewidth less than $k$ to a so-called d-SDNNF circuit, in linear time and with the dependency on $k$ being singly exponential.

The so-called d-SDNNFs are circuits satisfying conditions on the use of negation (only at the leaves), determinism (the inputs to OR-gates are mutually exclusive), decomposability (the inputs to AND-gates depend on disjoint sets of variables), and stucturedness (the AND-gates split the variables in some fixed way throughout the circuit, as described by a v-tree). This class has been studied in knowledge compilation and is known to enjoy tractable SAT and tractable model counting (recapturing probabilistic evaluation and counting), but other problems have been studied for this class such as enumeration, quantification, etc.

So one way to use bounds on the treewidth of a circuit is to convert it to this d-SDNNF class which has more explicit properties in terms of the circuit semantics, and for which there are several known results on the tractability of various tasks.