How does a private, permissioned blockchain exist? What is the incentive for a miner if they aren't rewarded with anything?
Surely at that point it'd just be a centralised database.
In a private blockchain, miners aren't needed. This is because all nodes in the private network are known and trusted. Users of the private network create signed messages that are recorded in the blockchain. Any computational or storage costs are shared by the private blockchain users.
Think of a consortium of 100 banks seeking to reduce costs and increase efficiency by clearing transactions among themselves. (Right now banks use clearinghouses for this) Every bank has its own copy of the blockchain so it's not centralized. They communicate over a secure network (i.e. not the internet) and use their own servers for storage and processing.
Keep in mind that the private blockchain I described is not Bitcoin but probably some kind of smart contracts like Ethereum.