There are more than just two parties to plagiarism. In the example given, the student buying an essay has not wronged the student who wrote the essay, anymore than a politician has wronged their speechwriter. However, when they turn in that essay for credit in a class, they have wronged the university by fraudulently representing it as theirs. The cost of the degree is in work that they have personally completed, not work that they have subcontracted out. This is the reputation that has been stolen, that has been inappropriately assigned to the plagiarist.
Where I would agree with your general statement is the concept of "self-plagiarism", typically defined as submitting the same work for credit in multiple forums. Plagiarism requires a third party whose work is being passed off, and self-plagiarism is an oxymoron.
Certainly the student has wronged the university, but to my mind it's the same kind of wrong as if they'd smuggled a calculator in to an arithmetic test. That also lets the student gain a reputation they didn't deserve.
I would prefer to reserve "plagiarism" for the case where the true originator of the ideas is potentially being eclipsed.
(Incidentally, according to my local university's academic misconduct policy, if the vice-chancellor uses a speechwriter without crediting them they are indeed a plagiarist. That doesn't seem entirely unreasonable to me, though I suspect it may not be enforced, but I would prefer to use a different term.)
Would it be better to deprecate the world plagiarism and fork it into the words cheating and reputation stealing? To me, that would ultimately be more accurate, because it better encapsulates the harm involved.
Where I would agree with your general statement is the concept of "self-plagiarism", typically defined as submitting the same work for credit in multiple forums. Plagiarism requires a third party whose work is being passed off, and self-plagiarism is an oxymoron.