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If bitcoin private key is no more safe with quantum computing and the public key of the private key is exposed, regardless of the wallet that you are using, the private key can be known as long as the public key is known. OP is very correct and adding cold wallets is off-topic tonehaybhenis discussing.
I don't suppose that the current quantum computers posses the capability to actually perform such a complex calculation yet. It'll only take a sufficiently powerful quantum computer to potentially use Shor's algorithm to factor the public key and derive the private keys of a cold wallet, and such quantum power hasn't been implemented yet, so using a cold wallet could still appear to be a safe way of storing Bitcoin in a post quantum world. Except I'm wrong...