We present a sample of 62 Hα detected broad-line active galactic nuclei (BLAGNs) at redshifts 3.5 < z < 6.8 using data from the CEERS and RUBIES surveys. We select these sources directly from JWST/NIRSpec G395M/F290LP spectra. We use a multistep pre-selection and Bayesian fitting to ensure a high-quality sample of sources with broad Balmer lines and narrow forbidden lines. We compute rest-frame ultraviolet and optical spectral slopes for these objects, and determine that 21 BLAGNs in our sample are also little red dots (LRDs). These LRD BLAGNs, when examined in aggregate, show broader Hα line profiles and a higher fraction of broad-to-narrow component Hα emission than non-LRD BLAGNs. We find that ∼90% of LRD BLAGNs are intrinsically reddened (βopt > 0), independent of contributions from emission lines to the broadband photometry. We construct the black hole (BH) mass function at 3.5 < z < 6 after computing robust completeness corrections. This BH mass function shows agreement with recent JWST-based BH mass functions, though we extend these earlier results to log (MBH/M⊙) < 7. The derived BH mass function is consistent with theoretical models, indicating that the observed abundance of BHs in the early Universe may not be discrepant with physically motivated predictions. The BH mass function shape resembles a largely featureless power law, suggesting that signatures from BH seeding have been lost by redshift z ∼ 5-6. Finally, we compute the BLAGN UV luminosity function and find agreement with JWST-detected BLAGN samples from recent works, finding that BLAGN hosts constitute ≲10% of the total observed UV luminosity at all but the brightest luminosities.