What needs to be written down for a paper wallet?
The design/structure of a paper wallet is not something is set in stone, but the main idea is writing public and private key pairs on a piece of paper.
The universal flow for generating an address is:

A public key is derived from private key
A bitcoin address is derived from public key

The common flow (Mnemonic) for generating an address is:

Master private key is derived from mnemonic phrases
A private key is derived from master private key
A public key is derived from private key
A bitcoin address is derived from public key

As you can see 3rd and 4th steps of Mnemonic key-generation flow are the same as universal key-generation flow. You can unlock your Bitcoin with plain private key without mnemonic phrases. Mnemonic phrases are just used to derive private key, so you can simply write down public key(65-byte or 33-byte) and private key(32-byte) onto a piece of paper regardless how you generated your key-pairs.