That's right. Mass, for example, is not a conserved quantity in all interactions. But energy is. That's why energy is a useful concept. It works as an accounting system, whereas mass doesn't.This is certainly a valid point of view if you believe in conservation of energy as if it were like counting beans. But we don't have an accounting system that is called "conservation of substance", nor is there a conservation law for matter the way there is for energy.
I am well aware, of course, of the sort of loose talk that says "mass is converted to energy in an atomic bomb" and things like that. That is strictly inaccurate. What happens there is that some mass simply disappears, and some other bits of mass end up moving faster than they did before. Also, some other particles appear - like photons, for example. To keep track of all this complexity, it is useful to keep track of an abstract quantity that is conserved in the entire interaction. Again, this is why energy is a useful concept.
The mistake that a lot of people make is to treat like some kind of substance. It isn't a substance. It's just a number. Richard Feynman, the Nobel prize-winning physicist, pointed that out in his famous Lectures on Physics.
I hope it is clear to you by now that I am saying that atoms are certainly not energy. Atoms are made of stuff; they aren't made of numbers. It's a category mistake to claim that anything is "made of energy". Regarding the "conversion" of matter to energy, see above.Your assertion that "atoms are not energy" is therefore about as inconsistent an accounting system as can possibly be. When matter converts to energy or vice versa, your accounting system doesn't work very well. What kind of science do you claim this is?
$$E=mc^2$$ just tells us what particular number to associate with a given amount of rest mass. To think that it is telling you that mass is "made of energy" is a category mistake.And as Farsight keeps pointing out, E=mc^2 works just fine as an accounting system to replace yours.