Can one manually type the mined block on bitcoin core and broadcast it?
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.