As a musician, acquainted with a technical, inclusive definition of "melody," I'm perfectly capable of peceiving that there is melody in Schoenberg's 12-tone works. The opening of his violin concerto - take, say, the first two minutes - has melody:
You can sing that in the shower. You can sing any succession of pitches in the shower, if you can remember them. I very much doubt, though, whether most people would be capable of remembering a succession of pitches like that. But they could probably remember on very brief acquaintance a melody like this:
or this:
or, with a little more difficulty, this:
It shouldn't be hard to hear what makes the Mendelssohn, the Sibelius, and the Bartok more memorable than the Schoenberg, though most people without musical training wouldn't be able to explain it. The question is whether most of us, who rely on our natural sense of what makes pitches hang together in something we recognize as a melody - our feeling for structure, internal relationships, balance, things having a beginning, a middle, and an end, progression toward a goal, pitches relating to an underlying harmonic plan and following each other in a way that feels logical and right - are talking nonsense when we say that the Mendelssohn, Sibelius and Bartok are melodious and the Schoenberg is not.
Technically, what the violin is playing in the Schoenberg is "the melody." But how well does it exemplify those qualities that make for clear, strong, memorable musical entities that make people say "That's a beautiful (or a charming, powerful, or haunting) melody," and have them singing in the shower without even realizing they're doing it? Pretty poorly, I'd say. Alongside that, arguments about the "correct definition" of melody are academic. One hears music, not definitions.