Effects of green tea tannin epigallocatechin-gallate (EGCG) on thermal-stress-induced amyloid fibril formation of reduced carboxymethylated bovine milk protein κ-casein were studied by dynamical light scattering and small angle X-ray scattering (SAXS). Two populations of aggregates, micelles, and fibrils dominated the time evolution of light scattering intensity and of effective hydrodynamic diameter. SAXS experiments allowed us to resolve micelles and fibrils so that the time dependence of the scattering profile revealed the structural evolution of the two populations. The low-Q scattering intensity prior to an expected increase in time due to fibril growth shows an intriguing rapid decrease, which is interpreted as the release of monomers from micelles. This phenomenon, observed both in the absence and in the presence of EGCG, indicates that under thermal stress free conditions, native monomers are converted to amyloid-prone monomers that do not form micelles. The consumption of free native monomers results in a release of native monomers from micelles because only native proteins participate in micelle-monomer (quasi)equilibrium. This release is reversible, indicating also that native-to-amyloid-prone monomer conversion is reversible as well. We show that EGCG does not bind to protein in fibrils, neither does it affect/prevent the proamyloid conversion of monomers. EGCG hinders the addition of monomers to growing fibrils. These facts allowed us to propose the kinetics model for EGCG-controlled amyloid aggregation of micellar proteins. Therein, we introduced the growth-rate inhibition function, which quantitatively accounts for the effect of EGCG on the fibril growth at any degree of thermal stress.