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Recently, it was shown that Kerr black holes can be described via the classical infinite-spin limit of a special class of scattering amplitudes in a massive higher-spin quantum field theory. Although this approach has successfully obtained state-of-the-art results for spinning black-hole binaries, only the three-point amplitude that describes Kerr is known in full generality and a full understanding of the underlying Lagrangian is still missing. In particular, vertices at four points and beyond are necessary to perform higher-order calculations. Massive higher-spin Lagrangians are highly constrained by properties such as unitarity and degrees-of-freedom counting. A useful tool in building consistent theories is the introduction of a massive gauge symmetry. However, constructing gauge-invariant vertices beyond the cubic level is a daunting task, so far never attempted in the literature. We propose an alternative on-shell realisation of gauge invariance, in the form of novel massive Ward identities, which provides a significant simplification with respect to the traditional approach. We show that the amplitudes known to describe Kerr are the unique lowest-derivative solution to the Ward identities combined with a known high-energy unitarity constraint. Moreover, we apply the same methods to compute new four-point Compton amplitudes for higher-spin states and propose them as candidates to describe higher-order black-hole observables. In parallel, we study the amplitudes of leading Regge states in superstring theory, as another example of consistent massive higher-spin particles. Applying the classical-limit formalism, previously only studied in the context of black holes, we recover known classical string solutions. This provides important insights on the properties of the formalism. Moreover, it paves the way to studying more general string states and attempting to reproduce black holes from strings.