I have to agree with you. In such a scenario if it were to happen it goes without say that burning would be preferable and the appropriate thing to do rather than allow them to be funnelled by quantum computers.
If it did come down to it I honestly cannot see anybody complaining about a hard fork if it was a simple choice between the end of Bitcoin or it carrying on (but those who did not move their coins before any fork just might have a differing view).
Every so often a possible threat to either Bitcoin or to private keys will emerge, Bitcoin and altcoins need to adapt to any perceived threat before they can cause any harm therefore need to be prepared for any and every eventuality.
Indeed. The question of what to do with the coins that are not moved to quantum-proof addresses is a huge problem.
From my amateurish perspective, it seems to me that if the problem couldn't be solved in time, and it came to a choice between either
(a) burning anything that hasn't been moved, or
(b) leaving them there to be scooped up by a QC
... then I think option (a) is far preferable.
You can't just soft-fork to a situation where some bitcoins are quantum resistant and some aren't; (b) could lead to another gox or worse.
A hard-fork option (a) would still be hugely contentious but if it comes down to a question of bitcoin's survival, it's the better option. Either way you're never going to get a consensus, and there would likely be a serious* chain-split.
*serious, not like BCH.