How does bitcoin core generate a receiving address without using my private key?
When the wallet initializes, it will generate a master private key for the you as part of the initialization. All subsequently generated keypairs will be related to the master private key, this is called a 'Hierarchical Deterministic' Wallet (HD wallet). 

An HD wallet is nice, because you only need to backup the seed phrase that generates your master private key, rather than backup every individual address.