I get that all miners compete to find the block first, which is then broadcasted to other nodes who then verify it. But what spikes my interest is would it be possible to broadcast the block to the blockchain without a miner's software's help? Can bitcoin core work alone for mining purposes?
Yes. Bitcoin Core has a RPC
submitblock, which, as the name implies, allows you to provide a block to be submitted to the Bitcoin network.
In fact, Bitcoin Core supports the
getblocktemplate protocol which means you can solo mine with Bitcoin Core. Furthermore, pools are probably running Bitcoin Core in their backend and their pool software just uses the
getblocktemplate protocol to get the transactions for a block and prepare the work that is sent out to the actual miners.