How is the underlying trust mechanism inefficient? You can run a fully validating Bitcoin node on an old laptop without any issues. Even a raspberry pi can handle it.
The computing power used for proof-of-work is not proportional to the number of transactions. Computationally, it's relatively cheap to verify the entire transaction history.
The real energy consumption comes from miners who use a crazy amount of computer power to secure the network. Bitcoin's transaction history is immutable because changing it would require an even crazier amount of computing power - too much for any malicious actor who tries to attack the network.
The amount of energy used is a feature, not a bug. There are no alternative designs known that could replace proof-of-work while keeping its security assurances, permissionless participation, fair distribution of coins, unforgeability, etc.
So if you had the world transacting on the blockchain ledger, it would take a crazy amount of computing power just to validate?