The short answer to your question is that positrons are not really electrons moving backward in time, and the premise of your argument doesn't work. However, something like what you are saying, is responsible for Hawking radiation.
There are a set of words you can give when you do QED in flat space along the lines of "positrons are electrons moving backward in time", but you really shouldn't take these words too seriously. Even ignoring gravity, electrons can be converted into neutrinos and quarks when you include the weak interactions (beta decay, inverse beta decay), so the notion that there is only one electron in the world that is jittering backward and forward in time every time a photon is emitted just doesn't work. The closest you can get is the CPT theorem... since quantum field theory is invariant under reversal of charge, parity, and time, then time reversal (T) is equivalent to reversing charge and parity (CP), which formally exchanges particles with antiparticles. But there's no way to actually implement a T transformation in reality. In controlled conditions, like a particle accelerator, you can set up an experiment done with one set of particles and the same experiment done with the CP-transformed particles to see what happens, and mathematically the results will be the same as if you had applied a T transformation, but at no point has time actually been reversed.
Now, there is something funny going on quantum mechanically with the fact that timelike and spacelike directions are switched beyond the event horizon of a black hole. But since we are sophisticated enough to realize that positrons are not electrons flowing back in time, we know it's not quite as simple as saying positrons will flow out of the event horizon. If you work through the math, you will find the implication of the horizon is that there are modes which have a negative frequency with respect to an observer at infinity. Performing a Bogoliubov transformation, this means that the state that looks like a vacuum to an observer near the event horizon, will look like it has particles to an observer at infinity. This is in fact Hawking radiation, and (in very crude terms) this is the strategy Hawking used to discover Hawking radiation in his original paper on the subject.
If you are really insistent, you can use these words to very crudely describe Hawking radiation: "particle-antiparticle pairs pop into and out of existence in the quantum vacuum, and near the event horizon sometimes a particle will escape the black hole and an anti-particle will fall in, or vice versa." You can loosely map these words onto the positive and negative frequency modes. But, like with "positron = electron moving backward in time" or "Feynman diagrams show trajectories of particles in spacetime," I would treat this more as a colorful analogy, than a rigorous description of what the math really says is happening.
As pointed out in the comments by @ChiralAnamoly, while I've phrased the answer in terms of the role of a timelike and spacelike coordinate switching roles at the horizon, physics cannot depend on your choice of coordinates. The coordinate picture in my answer is (I would argue) a fairly intuitive way of understanding what is weird about a black hole horizon and why you get negative frequency modes near the horizon, leading to Hawking radiation, it can be misleading to rely too much on coordinates. A more abstract but also more invariant way to describe what is going on, is in terms of different quantum vacuums states. An observer at asymptotic infinity will identify a certain state that is the "natural" vacuum, given the observer's worldline. An observer near the horizon will also identify a natural vacuum state. However, these two vacuum states are not the same. Positive frequency modes with respect to the horizon-observer's vacuum state, will be a mix of positive and negative frequency modes with respect to the asymptotic observer's vacuum state. This mixing gives rise to particle creation, aka Hawking radiation. The vacuum states for each observer are coordinate-invariant, as is the statement about the mixing positive and negative frequency modes (although the construction of the state is most easily done using coordinates adapted to the time of each observer).