  We continue the study of graph classes in which the treewidth can only be
large due to the presence of a large clique, and, more specifically, of graph
classes with bounded tree-independence number. In [Dallard, Milani\v{c}, and
\v{S}torgel, Treewidth versus clique number. {II}. Tree-independence number,
2022], it was shown that the Maximum Weight Independent Packing problem, which
is a common generalization of the Independent Set and Induced Matching
problems, can be solved in polynomial time provided that the input graph is
given along with a tree decomposition with bounded independence number. We
provide further examples of algorithmic problems that can be solved in
polynomial time under this assumption. This includes, for all even positive
integers $d$, the problem of packing subgraphs at distance at least $d$
(generalizing the Maximum Weight Independent Packing problem) and the problem
of finding a large induced sparse subgraph satisfying an arbitrary but fixed
property expressible in counting monadic second-order logic. As part of our
approach, we generalize some classical results on powers of chordal graphs to
the context of general graphs and their tree-independence numbers.
