Yes - you're missing "mathemagic" :) And a variety of independent engineering solutions which are largely independent of the mathemagic(think sharding or something like the lightning network).
Check out ZCash for an example of non-public, decentralized block chain.
- Transactions are on a decentralized blockchain
- That blockchain is however not public
- That blockchain allows for high(er) TPS
- That blockchain isn’t vulnerable to easy fraud
Am I missing a trade-off here?