UNCOVER spectroscopy confirms a surprising ubiquity of AGN in red galaxies at $z>5$
JWST is revealing a new population of dust-reddened broad-line active galactic nuclei (AGN) at redshifts $z\gtrsim5$. Here we present deep NIRSpec/Prism spectroscopy from the Cycle 1 Treasury program UNCOVER of 15 AGN candidates selected to be compact, with red continua in the rest-frame optical but with blue slopes in the UV. From NIRCam photometry alone, they could have been dominated by dusty star formation or AGN. Here we show that the majority of the compact red sources in UNCOVER are dust-reddened AGN: $60\%$ show definitive evidence for broad-line H$α$ with FWHM$\, >2000$ km/s, for $20\%$ current data are inconclusive, and $20\%$ are brown dwarf stars. We propose an updated photometric criterion to select red $z>5$ AGN that excludes brown dwarfs and is expected to yield $>80\%$ AGN. Remarkably, among all $z_{\rm phot}>5$ galaxies with F277W$-$F444W$>1$ in UNCOVER at least $33\%$ are AGN regardless of compactness, climbing to at least $80\%$ AGN for sources with F277W$-$F444W$>1.6$. The confirmed AGN have black hole masses of $10^7-10^9$ M$_{\odot}$. While their UV-luminosities ($-16>M_{\rm UV}>-20$ AB mag) are low compared to UV-selected AGN at these epochs, consistent with percent-level scattered AGN light or low levels of unobscured star formation, the inferred bolometric luminosities are typical of $10^7-10^9$ M$_{\odot}$ black holes radiating at $\sim 10-40\%$ of Eddington. The number densities are surprisingly high at $\sim10^{-5}$ Mpc$^{-3}$ mag$^{-1}$, 100 times more common than the faintest UV-selected quasars, while accounting for $\sim1\%$ of the UV-selected galaxies. While their UV-faintness suggest they may not contribute strongly to reionization, their ubiquity poses challenges to models of black hole growth.