This line alone shows you don't fundamentally understand the currency. The Cryptographic hash of the transaction block exhibiting specific, tunable properties (matching a number less than 'x'): was not chosen at whim. You should read the first reference in Shatoshi's Paper:
W. Dai, "b-money," (1998)It explains that you
can't do useful work while protecting the integrity of the system. The requirement to do "useful work" would allow dishonest participants to "cheat" by not actually doing the "useful" part of the computation.
incorrect. whether the computation is 'useful' is an externality; it has no direct bearing on its suitability for use in bitcoin.
a thought experiment: suppose computing sha-2 hashes below certain values were discovered tomorrow have a very important benefit to the progress of applied mathematics, or (more fancifully) to curing a disease or searching for extraterrestrial life. would that undermine the use of sha-2 in bitcoin? the answer is that it obviously wouldn't.
likewise, it is not possible to rule out more 'useful' substitutes for sha-2 hashing as proof-of-work - i.e., substitutes that have more positive externalities. some good analysts in the development thread have occasionally proposed alternatives (like the computation of mersenne primes) that seem at least potentially workable.
what a lot of people fail to understand is that mining is a
cost of bitcoin. it may be a necessary cost, but someone in the system has to pay for it. it is unclear in future whether the costs of bitcoin's mandated proof of work will outweigh the benefits of the currency. for example, the costs may be so great that the transaction fees that need to be paid to miners will be greater than the transaction fess of paypal or mastercard. that future comparison is, among the developers, an open question.
bitcoin was designed to be entirely decentralised, and decentralisation has substantial costs. it may also have substantial benefits. time will tell.