Is it an integral property of these blocks that brute-force is the accepted and preferred method? The intricacies of mining are a little over my head what with how little research I've done, but I would think that there ought to be a more efficient manner than brute-forcing, such as an algorithmic approach. Would such an approach inevitably introduce issues that brute-forcing wouldn't?
Yes. I recommend you read Satoshi white paper but blocks are SUPPOSE TO BE HARD. They are a "proof of work". If the work is easy/trivial then it isn't a proof of much is it. The blocks prevent double spending coins. Making the blocks difficult to produce means an attacker needs more computational power than all "the good guys" combined. The economics of mining mean that even if an attacker had that much computing power it would be more profitable to "do the right thing".
Obviously you could make producing blocks "easier" but the system would be easier to attack. Locating a solution by brute force has two major advantages. Solutions are found random. This prevents large miners from locking out smaller miners. If blocks were found in a deterministic manner then larger/faster miners would always win. Essentially the miner equivelent of a central bank. The second major advantage is that FINDING A SOLUTION is difficult but VERIFYING A SOLUTION is trivial. This means once a solution is found the rest of the network can verify it in a fraction of a second.
I strongly strongly strongly recommend everyone interested in the technical aspects of Bitcoin to read Satoshi paper. Once you read it (and read it again) you start to realize the elegance of the system. It is at the high level very simple and yet very robust.
