The PIN is the most important item stored on the hardware wallet, not the seed - because the seed is encrypted by the PIN (and by some other stuff I believe - I am not familiar with the internals of hardware wallets), but if the PIN is compromised, the seed might as well be stored in plain text at that point.
No no please... one thing that will never be stored on your hardware wallet is your pin. Your pin is used to encrypt the wallet and the recovery phrase and that encrypted recovery phrase will be stored on the wallet but not the pin.
It's a security measure to ensure that even if someone physically accesses your hardware wallet, they cannot use it without the PIN.
I find it confusing that when you create a hardware wallet, you are required to write down a 12-word "recovery phrase" in order to restore your wallet in the event that it is misplaced.
What makes this any different from simply leaving your wallet at home and writing down your private key on the same piece of paper?
I guess you've already received plenty of good answers. I just want to add that your recovery phrase is a key to your wallet, whether it's a hardware or a software wallet. If we say hardware wallets are used to store your "recovery seed/phrase," it is technically correct. However, even if your recovery phrase is not stored on the wallet, it will still function as intended. Your recovery phrase is stored on the wallet just for you, so you can retrieve it again in the future if required. Otherwise, a hardware wallet does not need your recovery phrase to be stored on the device after setting up or restoring your wallet.
So, yes two things are stored on wallets:
1. Encrypted recovery phrase for you to retrieve later.
2. Your public and private keys (used to sign transactions).
Everything in your wallet is locked (encrypted) using the "PIN."
What use does a hardware wallet serve if you have to write down what is effectively your private key anyhow?
Logically speaking, wallets are not supposed to physically store your recovery seed or phrase. Instead, they should generate your public and private keys based on that seed when you create your wallet. Your recovery seed is something you should write down and keep in a safe place that is not, or never connected to the internet.
That is the reason why every hardware or software wallet will recommend you to write down your recovery phrase on paper; this recovery phrase is your responsibility to be taken care of.
Now, to answer your question regarding if you write down your private key (recovery phrase) on paper, then what's the use of a hardware wallet? Can you use your recovery phrase to send transactions? No. You still need your private key to sign transactions, and that private key you will get after restoring your wallet. This is what a hardware wallet does for you. In this case, your private keys will never leave your hardware wallet, but you can still send transactions.
In summary, the recovery seed (or phrase) is your backup, and private keys are securely managed within the hardware wallet to sign transactions when needed.