I think when you reconstitute your private keys on a new cold wallet, your old wallet becomes obsolete and can be safely discarded. Your funds will now be stored on your new wallet, which is secured by your new private keys and passphrase.
Either way you look at it, this is incorrect. Coins are never stored in a wallet, only private keys are. If you are exporting private keys from your old wallet to your new one, then your private keys are only as secure as the weakest part of both your old wallet and your new one. The more wallets you expose your private keys to, the bigger the risk. If you are moving coins to new wallet via transactions, then your old wallet still isn't obsolete and shouldn't be discarded. You should keep a copy of it just in case.
Make sure to store your passphrase in a secure location, such as a password manager or a physical safe, to prevent it from falling into the wrong hands.
Bad idea to store your seed phrase or passphrase electronically, especially since most people use grossly insecure password managers such as those built in Google Chrome or something equally terrible.