Hawking radiation doesn't directly state that black holes can't exist, it predicts that black holes will lose more mass than it will gain and eventually disappear.
Right, but logistically, if you apply that to a black hole, where time is, compared to us, nearly at a stand-still, Hawking Radiation would affect the black hole incredibly fast (compared to us, again). So, essentially, black holes CAN exist, and probably do, but in our world, they won't just float there, they'll be rapidly (to us) diminished by Hawking Radiation in minutes or seconds, or fractions of seconds, because of how time moves inside of the hole.
~D-Draw