String-localized QFT allows to explain Standard Model interactions in an autonomous way, committed to quantum principles rather than a "gauge principle", thus avoiding an indefinite state space and compensating ghosts. The resulting perturbative scattering matrix is known (at tree-level) to be insensitive to the non-locality of the auxiliary "string-localized fields" used in the construction. For the examples of Yang-Mills and QCD, we prove that it is actually equivalent to the perturbative S-matrix of gauge theory, restricted to physical particle states. The role of classical gauge invariance is revealed along the way. The main tool are "dressed fields", that are intermediate between free fields and interacting fields, and for which we give explicit formulas at all orders. The renormalization of loops, as well as non-perturbative issues are not adressed, but we hint at the possibility, enabled by our approach, that qualitative traces of confinement may be visible already at the level of the dressed fields.