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Surprisingly good question, i would say. "yes" and "no"
Yes, physics specifically looks out to find "conserved quantities" and then goes on to ascribe great value to them, so, in some sense, yes, energy is considered important because it is a conserved quantity.
Also, yes, there are different types of energy (mechanical energy, light energy, chemical energy), and summing them all up via "energy" is indeed a bit of an accounting trick. but it's also a so fundamentally useful trick that we consider it "real".
Also, different than money, there is no central bank that can cause an arbitrary amount and cause an enormous inflation overnight. Similar to wood or cereals, there just is a limited supply of it per year, and that's what we can use. Physics makes the rules here.
Great write up.
Agreed, the money analogy breaks down when you bring in the cost fluctuation of a good. The whole purpose of energy is that the same conditions result in the same energy calculation.