We consider a simple model in which a non-elementary electroweak neutral Higgs scalar (H) can arise primarily as a multibody bound state which can include large numbers of constituent particles. Only standard-model elementary particles and effects with an energy scale have to be introduced explicitly. We do this by working with a non-perturbative analytic scattering-amplitude unitarization scheme (basically a relativistic generalization of the usual Schrödinger equation), but with a strong high-energy (multibody) inelastic contribution added in, and with no divergences requiring arbitrary cut-offs or subtraction constants. The H itself is required to be simultaneously consistent in both mass and coupling with the exchange of the same H in the crossed channel when constructing the relativistic generalization of the input potential. The strong energy dependence of our amplitude near the threshold then leads to a bound-state H with standard-model coupling and mass calculated to be near that threshold, a result which is expected to persist for a broad class of similar models and suggests that alternative solutions may be obtained with H immediately below the ZZ or thresholds. We also find that our high-energy scattering-amplitude multibody inelastic contribution can be constructed explicitly from the exchange of a Regge trajectory passing through the H and generated dynamically in the same way (but for unphysical angular momentum). Finally, we find that we can have massive and Z with longitudinal polarizations and approximately correct masses and standard-model couplings. These may be accompanied by higher-mass states with the same quantum numbers.