excellent explanation of a private key!!
But then again, what is the wallet if its now the public address?
The wallet can be seen as a container for your private keys and the derived public keys, and this container has special locks to open it (seeds, passwords, ...). Once you opened it, you can create with the wallet software a bitcoin transaction, that confirms to the rules of he network, and the wallet has software to send a transaction into the network. Also it is able to request data from the network (with light wallets), or it stores the blockchain itself on a local drive (full node).
So the wallet is a piece of software that makes it easy for the enduser, to communicate/participate in the network.