Do most wallets have both seed word and private key?
All wallets use private keys. If your wallet generates a seed phrase, it still generates private keys too, even if you don't see them. In some cases, such as Ledger devices as you mentioned, it is impossible to view the private keys since the device will not let them leave the secure element for safety reasons, although you can still derive manually from your seed phrase if you so desired.
Actually, a private key and a seed phrase have a lot in common: they are both a representation of a random number (entropy) that is used to generate public keys and addresses.
Important to note they are representations of different random numbers and they are not interchangeable.
For example, you can convert your private key to a seed phrase using the same BIP39 standard.
You can convert a private key (or anything else) to a phrase using the BIP39 wordlist, but it will not a be a seed phrase nor will it be following the BIP39 standard.