...and people will only do if the work accomplished is more valuable than the energy consumed.[sic]
I'm sure that you will agree that this is not true.
The only situation in which this is not true is if people are stealing their electricity, which is not a problem that Bitcoin can address. People who mine when it is not profitable do so because they consider helping the network to be more valuable than their electricity, or they think Bitcoins are undervalued (which are really just two ways of looking at the same thing).
Furthermore, how is solving the Byzantine General's problem helpful (to the world)?
It's not (at least, not directly), and it's not supposed to be. It's helpful to Bitcoin, and that's all that matters. You seem to be extremely confused about what Bitcoin is or should be. It is not, nor is it intended to be, some kind of system to harvest computer power to save the world, or for any other arbitrary purpose. Bitcoin is a
currency and a
payment system for that currency, and the resources it consumes serve the specific purpose of securing the payment system from various attacks and preventing the currency from being counterfeited. That's it. No other purpose is, or ever will be, served by the mining process.
You might as well ask for bank vault doors that create cancer drugs. Even if it were possible, it doesn't even remotely make sense for bank vault doors to be designed that way. That's simply not their intended function.