There's a world of difference between those two situations. The existence of sophisticated QC doesn't make cracking keys instant. It's pretty much accepted at this point that Bitcoin's signature algorithm could be broken in a few years, perhaps ten years in a best case scenario. But it'll take much longer than that to develop QC that can do so in the time it takes a transaction to confirm.

Also, Bitcoin users don't expose public keys unless they're spending. Until they spend from an address, the only thing exposed is a hash of the public key. So we're mainly worried about people who reuse addresses and people who have exposed public keys (to third party services or via Pay-to-IP which was removed from the reference client several years back).

The development of quantum computers is very slow so we have more time than that to implement algorithms that are resistant to quantum computers, so you are worrying for something that is not going to happen, it is very easy to think that something can destroy bitcoin but 10 years have passed and governments have been unable to find anything that comes close to that.