BANYAN. VII. A New Population of Young Substellar Candidate Members of Nearby Moving Groups from the BASS Survey

[Abbreviated] We present the results of a near-infrared (NIR) spectroscopic follow-up survey of 182 M4-L7 low-mass stars and brown dwarfs (BDs) from the BANYAN All-Sky Survey (BASS) for candidate members of nearby, young moving groups (YMGs). We confirm signs of low-gravity for 42 new BD discoveries with estimated masses between 8-75 $M_{Jup}$ and identify previously unrecognized signs of low gravity for 24 known BDs. This allows us to refine the fraction of low-gravity dwarfs in the high-probability BASS sample to $\sim$82%. We use this unique sample of 66 young BDs, supplemented with 22 young BDs from the literature, to construct new empirical NIR absolute magnitude and color sequences for low-gravity BDs. We obtain a spectroscopic confirmation of low-gravity for 2MASS J14252798-3650229, which is a new $\sim$27 $M_{Jup}$, L4 $\gamma$ bona fide member of AB Doradus. We identify a total of 19 new low-gravity candidate members of YMGs with estimated masses below 13 $M_{Jup}$, seven of which have kinematically estimated distances within 40 pc. These objects will be valuable benchmarks for a detailed atmospheric characterization of planetary-mass objects with the next generation of instruments. We find 16 strong candidate members of the Tucana-Horologium association with estimated masses between 12.5-14 $M_{Jup}$, a regime where our study was particularly sensitive. This would indicate that for this association there is at least one isolated object in this mass range for every $17.5_{-5.0}^{+6.6}$ main-sequence stellar member, a number significantly higher than expected based on standard log-normal initial mass function, however in the absence of radial velocity and parallax measurements for all of them, it is likely that this over-density is caused by a number of young interlopers from other moving groups. We identify 12 new L0-L5 field BDs, seven of which display peculiar properties.