The complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, $β$-acyclicity and bounded (incidence) MIM-width, are incomparable and lead to very different algorithms. We introduce the framework of point decompositions for hypergraphs and use it to derive a new sufficient condition for the tractability of (structurally restricted) Max-CSPs, which generalises both bounded MIM-width and \b{eta}-acyclicity. On the way, we give a new characterisation of bounded MIM-width and discuss other hypergraph properties which are relevant to the complexity of Max-CSPs, such as $β$-hypertreewidth.