Concrete recycling to produce aggregates is crucial in reducing the demand for virgin materials in the construction industry, particularly for the most widely used building material—concrete. A potential solution to enhance the quality of recycled aggregates involves the removal of the adhered cement paste from their surfaces. In this study, samples of industrial demolished concrete were selectively separated into recycled sands and aggregates while removing the hydrated cement paste. The recycled materials were characterized to assess their suitability for structural concrete production. The behavior and underlying mechanisms of recycled sands and aggregates proved to be identical, irrespective of their size. Water absorption emerged as a key parameter for evaluating the purity and quality of these materials. The statistical analysis revealed that when the water absorption of recycled aggregates and sand falls below 5%, the compressive strength may, at most, decrease by 15% (97.5% confidence) at any replacement level. Consequently, they can entirely replace virgin materials without a significant negative impact on fresh or hardened concrete properties at a constant cement content. Concrete samples with recycled materials had comparable workability, strength, and durability characteristics to references with virgin materials. The production of high-quality recycled aggregates and sand can facilitate complete concrete recycling and hence significantly contribute to virgin materials preservation, thus making concrete more sustainable.