how is that resolved for other things that radiate “thermally”?

For something other than a black hole, quantum field theory provides a fundamental description of everything that happens, and yes, you could track the time evolution for an individual quantum state and see that the end result is not truly thermal in its details.

But Hawking evaporation lacked a microscopic description. Lots of matter falls into a small spatial volume; an event horizon forms. Inside the horizon, everything just keeps falling together and collapses into a singularity. Outside the horizon, over long periods of time the horizon shrinks away to nothing as Hawking radiation leaks out. But you only have a semiclassical description of the latter process.

The best candidate explanation is the "fuzzball" theory, which says that singularities, and even event horizons, do not exist in individual quantum states. A "black hole" is actually a big ball of string which extends out to where the event horizon is located in the classical theory. This ball of string has a temperature, its parts are in motion, and they can eventually shake loose and radiate away. But the phase space of a fuzzball is huge, which is why it has a high entropy, and why it takes exponentially long for the fuzzball to get into a state in which one part is moving violently enough to be ejected.

That's the concept, and there's been steady progress in realizing the concept. For example, this paper describes Hawking radiation from a specific fuzzball state. One thing about black hole calculations in string theory is that they reproduce semiclassical predictions for a quantum black hole in very technical ways. You'll have all the extra fields that come with string theory, all the details of a particular black hole in a particular string vacuum, lots of algebra, and then you get back the result that you expected semiclassically. The fact that hard complicated calculations give you what you expect suggests that there is some truth here, but there also seems to be some further insight lacking, which would compactly explain why they work.

Here's a talk about fuzzballs.

nobody seems to mention the entropy carried by the radiation released during accretion

The entropy of the collapsing object jumps enormously once the event horizon forms. Any entropy lost before that is just a detail.

From a string-theory perspective, the explanation of the jump in entropy would be something like this: In string theory, you have branes, and then strings between branes. Suppose you have a collection of point-branes ("D0-branes") which are all far apart in space. In principle, string modes exist connecting any two of these branes, but in practice, the energy required to excite the long-range connections is enormous, so the only fluctuations of any significance will be strings that start and end on the same brane.

However, once the 0-branes are all close to each other, the energy required to excite an inter-brane string mode becomes much less. Energy can now move into these formerly unoccupied modes, so instead of having just N possibilities (N the number of branes), you now have N^2 (a string can start on any brane and end on any other brane). The number of dynamically accessible states increases dramatically, and thus so does the entropy.