No, your assumption is wrong.

All vanilla black holes (no charge, no spin) have an event horizon. If you are outside the event horizon, you are far enough away that light pointed directly away can escape to infinity. As in all gravitational wells, it will be red shifted. The light that reaches infinity will be less energetic that it was. light emitted very close to the event horizon is strongly red shifted.

Time passes slowly in a gravitational well. The deeper in, the more slowly. So it takes longer to escape than you might expect.

Inside the event horizon, everything, light included, travels on a trajectory that ends at the singularity. Light pointed outward does not travel outward.

Exactly at the event horizon, light would never go anywhere. Of course, you can't put light exactly at a point. It would either be a little inside and get sucked in. Or a little outside and an extremely red shifted remnant would escape.

No, your assumption is wrong.

All vanilla black holes (no charge, no spin) have an event horizon. If you are outside the event horizon, you are far enough away that light pointed directly away can escape to infinity. As in all gravitational wells, it will be red shifted. The light that reaches infinity will be less energetic that it would have been without the gravitational well. Light emitted very close to the event horizon is strongly red shifted.

Time passes slowly in a gravitational well. The deeper in, the more slowly. So it takes longer to escape than you might expect.

Inside the event horizon, everything, light included, travels on a trajectory that ends at the singularity.

Outside the black hole, there is a direction perpendicular to the event horizon toward the outside. An excited atom that decays can send a photon in that direction. It will travel infinitely far away. Photons sent in other directions travel on "straight" geodesics. But in curved space, "straight" bends toward the event horizon. The trajectory may well pass through the event horizon and end on the singularity.

However, if the excited atom falls past the event horizon, all directions lead to the singularity. Light pointed in what would have been the outward direction does not travel outward. Inside the event horizon, space is curved enough that intuition is not helpful. Perhaps you might think of the photon swimming upstream, but not fast enough. But of course there are problems with that thought.

Exactly at the event horizon, light would never go anywhere. Of course, you can't put light exactly at a point. It would either be a little inside and get sucked in. Or a little outside and an extremely red shifted remnant would escape.