If we are going with the substance/accidents distinction (which, I agree, could very much use an update/modern equivalent), then it is actually incorrect to say that a thing is, at minimum, its accidents. It has its accidents, but they are not essential to what it is.
And yes, in substance/accidents terms, even DNA and molecular structure (or, at least, the way we perceive them with senses or instruments) are accidental, not substantial. Substance is the underlying “thingness,” not any of the empirical aspects of the thing. That’s certainly weird — in all normal cases, the substance of a thing matches its accidents, to the point that most modern ontology doesn’t talk about substance as a distinct thing. If I recall correctly, even Aristotle would have been irritated at the use of his terms to apply to a special case in which substance does not match accidents; one of his big differences with Plato was his denial of ideal thingness as existing independently of individual examples of the thing.
What “goes away” in transubstantiation is the invisible quality of “breadness” and “wineness,” replaced by “Jesusness” without changing any of the sensible or measurable properties of the elements. Obviously this is not a scientific claim, or something intended to put forth evidence in favor of Christianity’s supernatural claims. Even before the science of atoms and molecules was fully understood, the theologians talking about transubstantiation knew they weren’t talking about something that could be detected or demonstrated. It’s something for us, inside the belief system. The reason for responding to the OP is not to present empirical evidence for the occurrence, but merely to dispel the notion that it’s not even logically coherent.
As others have noted, what goes on is a mystery for which transubstantiation is merely the best explanation we have so far. Jesus just said, “This is my body/blood,” and the majority understanding for most of the history of Christianity was that He did not mean that merely symbolically, even though there is no sign of change (though I don’t think anyone would deny that there is symbolic value to the use of a solid staple food for flesh and red liquid for blood). When the substance/accidents distinction of some early philosophers became available to Christian thinkers in the West centuries later, some bright spark thought, “Hey, what if what goes on at the consecration can be described as the one special case in which underlying substance is swapped out without changing the accidents ‘above’ it?” The less analytical East, meanwhile, is still like, “Enh, it happens. Who cares if we can put words to exactly how?”
