A multitude of JWST studies reveal a surprising over-abundance of over-massive accreting super-massive black holes (SMBHs) -- leading to a deepening tension between theory and observation in the first billion years of cosmic time. Across X-ray to infrared wavelengths, models built off of pre-JWST predictions fail to easily reproduce observed AGN signatures (or lack thereof), driving uncertainty around the true nature of these sources. Using a sample of JWST AGN identified via their broadened Halpha emission and covered by the deepest X-ray surveys, we find neither any measurable X-ray emission nor any detection of high-ionization emission lines frequently associated with accreting SMBHs. We propose that these sources are accreting at or beyond the Eddington limit, which reduces the need for efficient production of heavy SMBH seeds at cosmic dawn. Using a theoretical model of super-Eddington accretion, we can produce the observed relative dearth of both X-ray and ultraviolet emission, as well as the high Balmer decrements, without the need for significant dust attenuation. This work indicates that super-Eddington accretion is easily achieved through-out the early Universe, and further study is required to determine what environments are required to trigger this mode of black hole growth.