Where is the block reward system coded in Bitcoin core?
The coinbase transaction is added to a block template by a miner:

https://github.com/bitcoin/bitcoin/blob/4af01b37d40246cd1fdb54719855927e36a36b46/src/miner.cpp#L157-L167

When the block is solved, the entire block is broadcast to the network.

Outside of a new block, you'll never see a subsidy transaction being broadcast on their own. Here's the logic that rejects subsidy transactions from mempool acceptance:

https://github.com/bitcoin/bitcoin/blob/4af01b37d40246cd1fdb54719855927e36a36b46/src/validation.cpp#L575-L577