the problem that bitcoin "solves" is mathematically unsolvable. The only reason it kind of works is because participants are human, and therefore are able to assign arbitrary value to the currency, and therefore can act greedily to try to maximize (and protect) their coins. Participants are only incentivized to participate in mining because the thing they're rewarded with is a "currency" (something they value, as humans).

For anything but a currency, what is the incentive of miners using their resources to handle your transactions?