Why doesn't the Bitcoin core team develop official Bitcoin mining software? If I am right, the Bitcoin Core just maintains the software necessary to run a full node. Essentially it's the code for the Bitcoin wallet, which just verifies blocks, but doesn't mine. But mining is an important part of the Bitcoin network, and the most widely used software is the unofficial BFG miner written by Luke DashJr.

So why is it that the Bitcoin Core team does not develop an official Bitcoin miner as well? Also I believe the first version of Bitcoin core released by Satoshi was full node software as well? How did mining happen then?
Bitcoin Core does have mining components in it. In fact most available mining software actually rely on Core (or something abstracting Core's services) in order to get the block built before it attempts to mine it. Bitcoin Core has the capability to serve the information to other software so that they can mine blocks. Furthermore, Bitcoin Core does have the logic necessary to mine blocks. However it only does this via CPU mining, and because CPU mining is no longer viable, this has been disabled on the main Bitcoin network. It is still there in order to mine blocks on testnet and in the regtest networks.

Having a fully fledged mining software (i.e. one that supports all of the various ASIC miners and can connect to mining pools) is outside of the scope of the project. This would enable functionality that the vast majority of Bitcoin Core users never even use and thus would be a waste of time to develop.

Lastly, there is no official node software. There is no official anything in Bitcoin; it is decentralized, official would mean that there is some central authority dictating what is "official" and not.