If QC can break SHA256 is should be able to break most crypto so that's kind of doomish...
My prediction is that unless there is some kind of measure taken to prevent quantum computers stealing old and "lost" coins, we can pretty much expect that all of these will be "recovered" or "stolen" (depending your perspective).
In that scenario all of Bitcoin is toast. As is the entirety of the internet in its present form. Let's hope something is done before this becomes a thing.
Quantum computers recovering (or stealing) lost coins that are kept in old wallets does not rise to any kind of level of doom for the whole internet.
If old coins are not moved, and kept in old wallets, those wallets become vulnerable to quantum computers, but newer wallets can take security measures in order that they are not vulnerable to the same quantum computer attacks.
There could be some kind of measures to protect the old non moving coins from such attacks, yet I think that if in a worser case scenario they are recovered (or stolen) by someone who has such quantum computing capabilities, there is no real end to bitcoin because at such point soon thereafter, those coins would be put into circulation and potentially just dilute the bitcoin market for a short period of time, and price will adjust to such dilution.