There are two aspects to be addressed here.
Firstly it is not true that Hawking radiation is exactly always produced at the Event horizon. Usually for a field theory, we have vacuum fluctuations where there is creation and annihilation of a particle antiparticle pair. Now since the black hole has an event horizon, one particle can fall into inside the event horizon and the other escapes from the horizon, thereby reducing the total energy of the system in question. The thing is, these processes usually happen near the horizon, but not always.
Secondly, since the radius of the horizon keeps shrinking continuously due to Hawking radiation, therefore if a particle was very highly red-shifted initially, if won't be so sometime later. Given a black hole of mass M, you can calculate that it radiates its mass away in $t \approx M^3$, since the emitted Hawking radiation is blackbody radiation
The textbook understanding of Hawking radiation involves Bogoliubov transformations. The idea is that when you quantize lets say the electromagnetic field you take solutions of the classical equations (Maxwell's equations) and write them as a linear combination of positive-frequency and negative-frequency parts. Roughly speaking, one gives you particles and the other gives you antiparticles. Thus the way we choose the vacuum in our theory plays a strong role in the number of particles and the antiparticles. Outside the event horizon, the choice of vacuum is different and as a result we see a Hawking radiation due to particle production.