We continue the study of $(\mathrm{tw},ω)$-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this property has useful algorithmic implications for the Independent Set and related problems. In the previous paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. II. Tree-independence number], we introduced the tree-independence number, a min-max graph invariant related to tree decompositions. Bounded tree-independence number implies both $(\mathrm{tw},ω)$-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Set problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In this paper, we consider six graph containment relations and for each of them characterize the graphs $H$ for which any graph excluding $H$ with respect to the relation admits a tree decomposition with bounded independence number. The induced minor relation is of particular interest: we show that excluding either a $K_5$ minus an edge or the $4$-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most $3$ vertices, while excluding a complete bipartite graph $K_{2,q}$ implies the existence of a tree decomposition with independence number at most $2(q-1)$. Our constructive proofs are obtained using a variety of tools, including $\ell$-refined tree decompositions, SPQR trees, and potential maximal cliques. They imply polynomial-time algorithms for the Independent Set and related problems in an infinite family of graph classes; in particular, the results apply to the class of $1$-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius from 2019.