When I withdraw coin from a custodial service, I now control my key and the custodial service loses that control, so that if they were hacked, I would be safe because I am taking responsibility for security. So having a key that one writes down is immaterial. It's about who controls the key function.
This is essentially correct, but be aware it isn't that you are moving control of a private key from the custodial service to you, but rather the coins are moving from one private key to another, as denoted by a transaction in which the coins move from one address to another.
If someone was to give you the private key to an address with coins on it, then you have absolutely no way of verifying that they aren't also keeping a copy of the private key themselves. The only way to be sure that you and you alone are in control of the coins is to have them sent to a brand new address derived from a private key you have generated yourself in secret. A hardware wallet such as a Trezor does this for you, with every key it generates being deterministically derived from your seed phrase.
While this is possible, you shouldn't ever do this with a wallet which has coins on it. Typing your seed phrase in to a website is a fast track to having your wallet emptied and all your coins stolen. There is no need to view or write down individual private keys, and in fact, you are safer if you don't. As long as you have your seed phrase, you don't share it with anyone, and your coins are on addresses which are derived from your seed phrase, then you are firmly in control of your own keys and your own coins.