The issue of forming a reliable and sustainable structure of crumb-rubber-modified binder is an important scientific and technical task. The quality of this task will increase the technical and economic efficiencies of road construction materials. This work is dedicated to developing a scientifically justified method of directed thermomechanical devulcanization, which ensures the solubility of the crumb rubber in the complex structure of a polydisperse composite material, preventing the formation of aggregates consisting of unsaturated crumb rubber particles, whose elastic aftereffect causes intensive cracking, especially during low-temperature road operations. The novelty in the first part of this article is due to the fact that, for the first time, the quantitative ratio of the polymer component in the crumb rubber was experimentally determined. The ratio of the polymer component to the total content of the other rubber components in the crumb rubber (CR) was determined to be, on average, 93.3 ± 1.8%. The stabilities of the compositions of crumb rubber from different batches were experimentally studied. The nature of the polymer component in the crumb rubber was determined. A hypothesis was formulated to obtain a thermodynamically stable and sustainable binder modified with crumb rubber. To evaluate the compatibility of hydrocarbon plasticizers with the studied CR samples, the following semi-empirical and thermodynamic compatibility parameters were calculated: Hildebrand solubility parameters based on evaporation energy and surface tension, Barstein’s compatibility parameter |X|, Traxler coefficient, and the mass ratio of paraffin naphthene:asphaltenes. It was shown that for the substances under study, it is advisable to justify the choice of plasticizer based on chemical compatibility criteria. It was established that a supramolecular plasticization mechanism occurs in the “hydrocarbon plasticizer–crumb rubber” systems under consideration. In the development of the crumb-rubber-modified binder, it was found that the use of activated crumb rubber (ACR) from large tires does not ensure the achievement of a stable and resilient structure of the crumb-rubber-modified bitumen.