> signing every block, to create a collaborative system that is perfectly suited for the same task.
Indeed you can! We even have a name for that! Its called a blockchain.
> This maintains many benefits of the blockchain and lacks many issues (fast, simple, near zero cost, controllable to a given extent -- no takeover possible, ...).
Blockchains can do all of these things.
Perhaps you are thinking of "bitcoin", instead of "blockchains"? Bitcoin, something that was created a whole 17 years ago, indeed has many drawbacks compared to modern blockchains.