Miner’s pool and Mempool relationship

I am wondering why we need Mempool (central and aggregated pool) while each miner has its own pool?

If we need the Mempool as an architectural design, how we can make sure that miners are sending their transactions to Mempool?

Is that possible for a miner to confirm a transaction on its own(Privately) but not send it to the Mempool(Shared-Public Pool?

Please correct me if my understanding from Miner pool as a private and the Mempool as a public area is not correct.

Thank you