Recently, using X-ray free electron laser radiation, the first direct experimental evidence of the existence of a paired state of charge carriers in real space was obtained in the parent compound BaBiO₃ of the high-temperature superconductor (HTSC) family (bismuthates) having a perovskite-like structure similar to the structure of cuprate HTSCs. As a result, the main provisions of the previously formulated model of a spatially separated Fermi—Bose mixture were confirmed, in which a new original mechanism of high-temperature superconductivity in bismuthates was proposed. In this review, we present the development of this model based on the results obtained on an X-ray free-electron laser and consider in detail the complete phase diagram of the superconducting and normal states in bismuth oxides Ba_1-xK_xBiO₃ at various potassium doping concentrations. We discuss new unique quantum states of matter in the form of a bosonic insulator (semiconductor) with initially paired charge carriers and two energy gaps and a bosonic metal shunted by a fermionic component, and provide experimental evidence that it is precisely the local pairing of electrons and holes that is responsible for the set of basic anomalous properties of the bismuthate family. Considering the large number of analogies in the behavioral features of various families of perovskite superconductors, we believe that our work will give a new impetus to understanding the nature of high-temperature superconductivity in bismuth oxides and other families, including cuprate HTSCs.