We perform the first fully general relativistic, magnetohydrodynamic simulations of dynamically stable hypermassive neutron stars with and without ergoregions to assess the impact of ergoregions on launching magnetically driven outflows. The hypermassive neutron stars are modeled by a compressible and causal equation of state and are initially endowed with a dipolar magnetic field extending from the stellar interior into its exterior. We find that, after a few Alfvén times, magnetic field lines in the ergostar (star that contains ergoregions) and the normal star, have been tightly wound in both cases into a helical funnel within which matter begins to flow outward. The maximum Lorentz factor in the outflow is ΓL∼2.5, while the force-free parameter holds at B2/8πρ0≲10. These values are incompatible with highly relativistic, magnetically driven outflows (jets) and short γ-ray bursts. We compare these results with those of a spinning black hole surrounded by a magnetized, massless accretion disk that launches a bona fide jet. Our simulations suggest that the Blandford-Znajek mechanism for launching relativistic jets only operates when a black hole is present, though the Poynting luminosity in all cases is comparable. Therefore, one cannot distinguish a magnetized, accreting black hole from a magnetized hypermassive neutron star in the so-called mass-gap based solely on the value of the observed Poynting luminosity. These results complement our previous studies of supramassive remnants and suggest that it would be challenging for either normal neutron stars or ergostars in a hypermassive state to be the progenitors of short γ-ray bursts.