... In other words, energy can neither be created nor destroyed, and energy, in all of its forms, has mass...
Here is an example of one of my objections, to the reasoning involved in the whole quote.
First, let me say I originally began from essentially this same understanding. Over the last few years I have been attempting to understand some of the work done in explaining inertia and gravitation from the perspective of quantum mechanics. (quantum mechanics was not my favorite subject, so the process is less than ideal)
If you take the quote from your post literally, then one would have assume that the zero point energy of quantum mechanics, has mass. This is essentially saying that empty space where a ground state non-zero energy (ZPE) is present, has mass. While conceptually this could help to explain some difficult observations, it also raises significant issues. ZPE begins to resemble some form of ether that would not only act as a medium for the propagation of EM radiation, but may also interact dynamically with it.
If empty space has mass then an interaction between photons and empty space, become more difficult to explain, without some degradation of the photon.., this suggests a tired light potential and problems for our current understanding of a large portion of the observed cosmological redshift.
While from what I can understand of the quantum argument for the origin of inertia and gravitation, it does seem to me that "empty" space must be considered to be filled with some intrinsic substance or energy, I am not sure that whatever practical form that substance/energy may have or take, it is not likely one we could describe as mass, within the current context of the concept of mass.
There does appear to be a significant link or connection between what we understand of mass and energy, mass seems to be emergent rather than a fundamental component....
There is a second issue in that the quote itself suggests that all energy has mass, and we know with some degree of certainty that while gravitational and inertial mass are equivalent, gravitational mass does not increase with an object's velocity. Kinetic energy associated with relative motion does not add or subtract from an object's gravitational field or potential.
While there is a close relationship between mass and energy, and mass may in fact be an emergent phenomena dependent on energy, they are not equivalent and entirely interchangeable. Though mathematically this may be possible, as a matter of practical application mass and energy continue to have some independently defined characteristics.
Some of the discussion involves principles of thermodynamics originating from closed systems. Ideally it is convenient to project closed system observations, into the world generally. The problem is when we do so we must also define the expanded closed system. We cannot with any real certainty limit the world and the universe to a closed system, that can be locally defined.
In some respects this question may come to one of, while we can accept that $$E = mc^2$$ represents the energy content of a given mass, it does not exclude any contribution to that mass of anything other than energy. Though it does seem that, at least a significant portion of the theoretical approach, to this subject does attempt to suggest just that.
