We prove the continuity of the extremal decomposition measure of the free state of low temperature Potts models, and more generally of ferromagnetic finite-spin models, on a regular tree, including general clock models. The decomposition is supported on uncountably many inhomogeneous extremal states, that we call glassy states. The method of proof provides explicit concentration bounds on branch overlaps, which play the role of an order parameter for typical extremals. The result extends to the counterpart of the free state (called central state) in a wide range of models which have no symmetry, allowing also the presence of sufficiently small field terms. Our work shows in particular that the decomposition of central states into uncountably many glassy states in finite-spin models on trees at low temperature is a generic phenomenon, and does not rely on symmetries of the Hamiltonian.