From what I understood about tampering into public blockchain like Bitcoin or Ethereum:

We cannot just add a transaction into an existing block because changing one would mean changing its hash, and that would mean it would have to be mined again.

A miner could try to do that, but in the meantime, all the other miners have moved on to the next block, so they will have a longer blockchain, and the longest blockchain is always the “correct” blockchain.

Now my question is,
What if the blockchain is private, there’s no incentive given for mining blocks, correspondingly there are no miners,

How will the new blocks be mined then? Is it done by a dedicated group of servers? And what if my computation power is larger than the computation power of those servers? Does it mean that my tampered blockchain can be longer than the blockchain mined by those servers and thus it can be accepted as “correct” blockchain?