Can we infer Hawking radiation assuming the Unruh effect?
I would say, cautiously, that we can at least infer some thermal properties of a black hole including its Hawking temperature from good understanding of Unruh effect, but the interpretation of this thermal effects as “radiation” coming from the black hole requires additional assumptions about the state of quantum fields in the region around the black hole.
An observer near the event horizon of a black hole will experience an extremely strong gravitational field.
I would suggest using more precise phrasing in such situations, distinguishing between “gravitational field” in the sense of spacetime curvature and acceleration or, more specifically, proper acceleration of observers as a measure of non-inertiality of their trajectories. The distinction between curvature as an observer independent measure of gravitational field and the acceleration can already be understood in Newtonian gravity. Acceleration at a given point could be made arbitrary by choosing appropriate non-inertial accelerating reference frame. On the other hand tidal effects are independent of local values of acceleration and are described by derivatives of Newtonian acceleration field: $\nabla_{(i} g_{j)} $ and so it is a true geometric, observer independent quantity. In GR this quantity is generalized to a curvature tensor field.
For an isolated body Newtonian acceleration behaves as $g\sim M / r^2$, so the components of curvature tensor behave like $\sim M / r^3$. This behavior of curvature remains true in full GR if we use coordinate independent measured of curvature (so called scalar curvature invariants, such as Kretschmann scalar) even across horizon. So the curvature at the horizon $\sim M/r_S^3$ remains bounded and since $r_S\sim M$ it decreases with increasing mass of the black hole. This is often emphasized by saying that local inertial observers do not notice anything special when passing through event horizon.
So, the curvature near event horizon can be small, if the black hole is heavy enough.
On the other hand the acceleration of an observer held static near the event horizon diverges as they approach the event horizon. Example calculation of such proper acceleration can be found in this answer.
This diverging acceleration and the Unruh effect that would be experienced by these observers can help us deduce black hole temperature, if we also recall general properties of thermodynamical equilibrium in static gravitational fields, namely the Tolman temperature gradients, spatial dependence of the equilibrium temperature on the redshift caused by static gravitational fields (see e.g. this answer for the basics and further references).
Now, assume that we have some static material structure around the (static) black hole. Some parts of that structure are very close to black hole event horizon, while other parts are far enough that the gravitational time dilation there is negligible. Then if we assume that there is a thermal equilibrium between the constituent parts of that structure and between black hole event horizon, then the redshifted temperature of that equilibrium (i.e. the temperature measured where gravitational redshift is negligible) would be the Hawking temperature of that black hole. Indeed, for parts/observers very close to the horizon, the temperature of thermal equilibrium is determined by Unruh effect alone, while the Tolman gradients would connect local acceleration dependent temperature of near-horizon observers with global redshift adjusted temperature of equilibrium.
While this considerations give us the Hawking temperature, the “radiation” part is another matter. The temperature is independent on the specific mechanism of achieving thermodynamical equilibrium, while the characteristics of the Hawking radiation is a much more difficult subject. It is possible to obtain some quantitative estimates for Hawking radiation from flat space properties of thermal radiation (such as Stefan–Boltzmann law), but generally we would need the full machinery of QFT in curved spacetime for details.
