Sure, so I'm not sure which "output" you mean. The output of the computationally-intensive part is really just a magic number that solves a puzzle. Once a miner finds that magic number, they publish a block that has the most recent transactions (which are all verified separately), as well as the special transaction where the miner grants themselves the current mining reward. Other miners see it and verify that all that stuff was correctly computed.
So that whole block (which is maybe what you're calling the "output"), does contain the puzzle solution as well as all the latest transactions. But the actual specifics of the puzzle (which is currently brute-force SHA1 reversing), aren't super important. It could always be swapped out for some other task.
By 'output' I (think) I meant the output we get for the energy used by the miners.
You seem to be saying that The Miners have to verify transactions to claim their reward? When you say "which are all verified separately" - you mean verified by the same miner who publishes the block? If so then the bitcoin network as a whole benefits from their energy use.
On the other hand, I see your point that the make-work is not intrinsically linked to transaction verification. But my understanding is, its the difficulty of the make-work that protects the block-chain from being hijacked by ..er... 'the bad guys', whomever they may be.
So, the whole thing kinda makes sense, and the 'make-work' is like an energy commitment that keeps the whole thing integrated. The assumption being, the total commitment of all bitcoin users will always be greater than the commitment of any single attacker/attacker group.
So that whole block (which is maybe what you're calling the "output"), does contain the puzzle solution as well as all the latest transactions. But the actual specifics of the puzzle (which is currently brute-force SHA1 reversing), aren't super important. It could always be swapped out for some other task.