To answer your question quickly: there are problems with the derivation of the Hawking effect and it does involve assuming validity of results that could depend on quantum gravity, but it is not due to the reasons you pointed out and we have strong belief that the result holds nevertheless (see pp. 12–13 of arXiv: gr-qc/9912119 and references therein, DOI: 10.1038/s41586-019-1241-0 should also be mentioned). See also Wald's 1994 book Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics, especially pp. 162–163.
As mentioned in the comments, a freely falling observer does not observe Hawking radiation (see, e.g., Wald's book on p.171). The thermal effects are observer dependent and they assume explicitly the observer to be following the orbits of the static Killing field. By studying the problem from the point of view of such an observer, no issue should occur.
To further explain why this argument holds, notice that while the static and the freely falling observer are locally related by a Lorentz transformation, these effects are not local. Locally, we could also relate an inertial observer and an accelerated observer in Minkowski spacetime by a Lorentz transformation. However, one of them perceives the Minkowski vacuum as a freezing void, while the other one perceives it as a hot mess of particles. It is not sufficient to use the local structure to determine the quantum effects. This can also be seen, for example, from the fact that the derivation of the Unruh effect employs a bifurcate Killing horizon, which is a global notion.
To avoid misinterpretation of the previous paragraph, I should also recall, as was done in the comments, that Hawking radiation is not the same as the Unruh effect. The Unruh effect can be generalized to curved spacetimes, but in the case of a black hole, it does not correspond to the Hawking effect. As pointed out, the Unruh radiation comes from all around a static observer in the presence of an eternal black hole. Hawking radiation, on the other hand, is seen by an static observer as coming from the black hole, which is assumed to be formed by gravitational collapse, instead of being eternal. Both effects are very well discussed on Wald's book, and the fact that they are quite different is pointed out, for example, on p. 129.
As for an observer-independent argument for the backreaction, I believe you would agree with me that the backreaction effects would only be divergent if the expectation value of the stress energy tensor would diverge as well. However, the Hawking effect is derived in the Unruh vacuum, which is well-behaved (more specifically, it is a Hadamard state) near the (future) event horizon, and hence no such an issue occurs. This vacuum is characterized by the fact that static observers see particles coming from the black hole with a thermal distribution at the Hawking temperature and see no particles coming from infinity. If one analyses the problem in this observer-independent manner, no issue could possibly arise, and the problem would only happen in the frame of a static observer, where it is natural since, as pointed in the comments, one already needs infinite acceleration to stay there. More information on this can also be found at Wald's book.
As you might guess, I quite like Wald's book hahaha. I believe all of these issues are treated there in clear way, especially on Chap. 7, which concerns the Hawking effect. While it is not an easy read, it is definitely worth the effort.
