These private mining datacenters typically get their block templates from pool servers. Miners communicate with the pool servers over the internet. Control the pool server control the mining power.

This is exactly how it works! Most miners are connected to pools like slushpool, f2pool, etc over unencrypted an unauthenticated links. If you can modify this traffic you can steal the hashrate, because you can modify the work being sent to the miners before they do any hashing.