You are correct: there is no Hawking radiation for a general spherical body. The reason is subtle, although the reference you gave comes near it on p. 17: the field's state.
In the reference you provided, the author decides that the physical choice of vacuum is $|0_K\rangle$ because $|0_T\rangle$ has an unphysical build up of modes near the event horizon. This is not seem, so $|0_T\rangle$ is not physical and must be abandoned.
However, suppose that we have a planet instead of a black hole. Then we have two remarks:
- $|0_T\rangle$ doesn't have an unphysical build up of energy near the horizon, for the horizon literally does not exist (the solution near the "would-be-horizon" is an interior solution, not Schwarzschild)
- $|0_K\rangle$ has unphysical modes that go into the horizon, which literally does not exist. Hence $|0_K\rangle$ is not an acceptable state anymore.
This is the point in which the derivation fails. The physical states in a black hole spacetime and in a planet/star spacetime are different. If the state is different, the behavior is different, and hence we end up with no thermal effects.
To give a few more nomenclature, here are three of the main states considered in Schwarzschild spacetime:
- Hartle–Hawking vacuum: this is the vacuum associated with the Unruh effect in curved spacetime. This vacuum is compatible with the symmetries of Schwarzschild spacetime and it has no unphysical build ups of energy. It predicts a static observer hovering above the black hole will see particles coming from the black hole and from infinity with a thermal spectrum.
- Unruh vacuum: this is the vacuum associated with the Hawking effect (yes, the Unruh vacuum is associated with the Hawking effect, and the Hartle–Hawking vacuum with the Unruh effect). This vacuum is compatible with the symmetries of Schwarzschild spacetime and it an unphysical build up of energy near the white hole horizon. This is not an issue if you consider a black hole that formed out of a stellar collapse, in which case the white hole itself is unphysical. It predicts a static observer hovering above the black hole will see particles coming from the black hole but not from infinity with a thermal spectrum.
- Boulware vacuum: this is the vacuum in which static observers hovering at a constant distance don't see any particles. It has unphysical build ups of energy near the event horizon (I believe both in the black and white holes, if my memory is not failing me). This, of course, is not an issue if the spacetime has a planet or star that renders the horizons unphysical. This vacuum does not predict a static observer will see particles with a thermal spectrum coming from either the spherical object or from infinity.
Notice that the Unruh effect and the Hawking effect are physically different phenomena (as pointed out in Wald's QFTCS book). It is very common, however, to derive the Unruh effect in Schwarzschild spacetime and call it the "Hawking effect", although Hawking's original derivation was concerned with a completely different field state and hence has different predictions concerning what is observed.
For more information, I suggest checking out Chap. 5 of Wald's book. Everything I said in this answer is probably there somewhere.