Suppose that in some sexually reproducing species, a perfect DNA-copying mechanism is invented. [...] The perfect DNA-copying mechanism rises in frequency to fixation. Ten thousand years later there's an ice age and the species goes out of business. It evolved to extinction.

This bit is surely not technically very likely. We have a range of species with different error-correction rates. It is pretty clear that the rate is largely under adaptive control - and that better and worse error correction mechanisms arise all the time.

High quality error correction has drawbacks - partly because mutation rate control requires energy and resources - and so it is not usually selectively favoured beyond a certain point.