If ECDSA is broken and it is possible to recover the private key from the public key, then even if we depreciate ECDSA signatures then there is nothing stopping an attacker still recovering the private key and also making a transaction on the new algorithm and stealing your coins. As discussed above, the only way around this would be to require a zero knowledge proof of some other piece of information such as the parent chain code or the seed phrase which derived that private key, but that would only be possible for addresses which were part of an HD wallet.

Rather, we would need to have a quantum resistant address type well in advance of ECDSA being broken, everyone would have to generate new quantum resistant addresses from new private keys, and then move their coins to these addresses prior to them becoming vulnerable.