Unveiling the Milky Way with a Gaia DR3 census of OB-type stars within 2 kpc. I. Tracing local Galactic structure, massive star-forming regions and core-collapse supernova progenitors

O- and B-type stars are young and hot, thereby serving as vital tracers of the star formation and spiral arm structure of the Milky Way. At the dusk of the Gaia DR3 era, a high-confidence and accurate catalogue appears timely. Here we have characterized a population of 105,971 OB-type stars (T eff > 10,000 K; hereafter OB stars) within 2 kpc from the Sun, using an astro-photometric Bayesian inference tool. Our resulting map unveils a complex view of the young stellar populations across the thin disk, with prominent large-scale features such as the Cepheus Spur, the Giant Oval Cavity, and a segment of the Sagittarius-Carina spiral arm all visible. Their inhomogeneous spatial distribution implies that massive star formation has taken place clustered across a few highly concentrated regions. We find a correlation between the overdensities of OB stars and young open clusters (< 20 Myr), although OB stars can be better detected in high-extinction regions. We identify over 4200 OB stars as core-collapse supernova (ccSN) or direct-collapse black hole (BH) progenitor candidates, and therefore targets of interest for spectroscopic follow-up. Furthermore, we find no OB-type star ccSN progenitor to explode within the next 1 Myr within 100 pc, at which such an event could be harmful to Earth's biosphere. Finally, we identify more BH progenitors to collapse within the next 1 Myr than ccSN to explode, despite the former's much scarcer number - which could be indicative of a recent massive star formation burst in the local Milky Way.