I'm not particularly satisfied with the answers that cite topology change or breakdown of semi-classicality. Particularly the latter doesn't explain why a metric which appears more and more flat seems to produce hotter and hotter Hawking radiation well before the Planckian regime is approached.

But I've thought about this for a while and finally have found a way to be at peace with this behaviour (save for the incompatibility of asymptotic flatness and a temperature at infinity that QGR pointed out, which I've posed as a [separate question][1]).

Here's how I'm thinking about it - do leave a comment if you disagree:

Close to the horizon, a stationary observer sees the Rindler metric and the Unruh temperature here is equal to the Hawking temperature - the observer has to fire their engines with a proper acceleration equal to the appropriate proper acceleration in the Unruh case. This also means that a free-falling observer sees no radiation at all (this is true at least close to the horizon, which I think is enough).

So, as I send the limit $ M \rightarrow 0 $, the appropriate observer that I'm mapping on to as I approach flat spacetime is *not* the inertial observer in flat spacetime, but the eternally accelerating observer with proper acceleration $ 1/M $. (Unruh proper acceleration is identified with the surface gravity)

So *of course* I should expect to see a thermal state of infinite temperature in this limit to flat spacetime since I'm not being mapped to an inertial observer in Minkowski, but to an eternally accelerating Rindler observer with infinite acceleration.

If I were always considering geodesic (free-falling/inertial) observers, at least close to the horizon, this limit makes perfect sense because there is no radiation (thermal or otherwise) during and after taking the limit.

PS: Note that after asking the question, I confused this limit taking process with black hole evaporation - which is not really what I was asking - I just wanted to know what the temperature of the spacetime looks for flatter and flatter spacetimes - and not the complicated stuff that happens when a black hole actually evaporates. Or in other words, I wanted to know what the $M=0$ element in the set of eternal black holes looks like, and what it looks like is the spacetime seen by a Rindler observer with infinite acceleration in flat spacetime.




  [1]: https://physics.stackexchange.com/q/10508/1228