A seed phrase is used to generate a master key and chain code for an HD wallet. These values can be used to generate many keys and addresses depending on how you want to set up your HD wallet.
There is no way to use any of the private keys to generate the master key or mnemonic. But the mnemonic can be used to regenerate the master key, chain code and any of the private or public keys.
You would never want to send crypto to a public key. You would send it to the address generated by the public key and it would appear in whichever wallet uses that key. An HD wallet can manage a very large number of addresses, and each one would have its own private and public key.
For more information, you could read about BIP 32, BIP 39 and BIP 44.