Modern astronomy is still clinging to the historical method of naming things according to their influence on and by their surroundings. Yes, it's silly. I'd rather a system that describes bodies by their characteristics instead of their locations relative to other objects.
Phobos is just an asteroid captured by Mars, but because it's a satellite of Mars, it is classified as a moon. Some large moons around gas giants like Triton are thought to have once been planets or dwarf planets. Obviously, there's a huge difference between Titan with its large diameter and thick atmosphere and the tiny, irregularly shaped Deimos. Yet, they are both moons. The current system cares nothing for their characteristics or how they were formed.
Your notion regarding the Earth/Moon system having a different classification in the future than it does today is no more relevant than if Mars were to be captured by Jupiter in a few billion years due to orbital instability. We'd then call Mars a moon instead of a planet. Odd, perhaps, but that doesn't mean we should start calling Mars a Jovian moon in the meantime - even if we had mathematical models proving it was going to happen. Even more odd -- if a Jovian moon were to be hurled into deep space, there is no official designation for what to call it at that point. Rogue planet, sub-brown dwarf, and interstellar planetary body are merely suggestions.
To say that the Earth and the Moon have a special relationship is obvious, but it doesn't warrant any extraordinary classification given the absurdity of the current system. The Moon does indeed orbit Earth as evidenced by the barycenter being inside Earth as well as the Earth being the more massive of the two objects. The Moon is also tidally locked to Earth as most moons are. Yes, as you point out, both the Earth and the Moon have a sinusoidal/elliptical orbit around Sol, but I'd argue that it's not only insignificant, but its shape would change entirely if Earth/Moon were a greater distance from Sol. Earth's orbit only shows a tiny wobble while the Moon's is more pronounced, but more importantly - the shape of the Moon's orbit has little to do with the mass of the moon itself. At its current distance from Earth, given the masses and positions of Sol and Earth, any satellite would have a stronger gravitational influence from Sol than from Earth. However, if we moved Earth and the Moon to a distance say... in place of Jupiter, Sol's influence would be less, the barycenter between Earth and the Moon would stay the same, but you'd instead see a true looping orbit path for the Moon around the Earth as Earth's gravity will be significantly stronger than the Sun's at that distance. Some of Neptune's outermost moons also have a sinusoidal path around the sun, but again, it's more to do with the masses and distances to Neptune and Sol than anything unique about the moons in relation to Neptune. Alternatively, over 4 billion years ago, the Earth/Moon system would have had a barycenter closer to Earth's core and the distance between the Earth and Moon was shorter, making Earth's gravitational effect on the Moon much stronger. I haven't done the math, but given the distances, I'd bet the Moon's orbit at that time was more influenced by Earth than Sol as well and took a different shape around Sol.
Your argument that the Earth and Moon have a special relationship in our solar system is valid, but your argument to classify that relationship as a binary planet is flawed primarily because there currently is no such formal classification. One was proposed for Pluto because the barycenter for it and its moons is outside of Pluto, but that proposal was abandoned. Pluto is a dwarf planet with moons instead of a binary planet or even a binary dwarf planet with Charon. The barycenter idea isn't a rule - it's just an arbitrary argument which wasn't strong enough to convince a committee that it was important enough to warrant such a distinction. There are other arguments such as the relative sizes of the planets, whether they formed from the same material or one captured another, the relative gravitational influences of the host planet and the star and so on. Nothing was convincing enough to warrant a separate designation for Earth/Moon or Pluto/Charon.
Sadly, since there's no such formal classification, even billions of years from now if the Earth/Moon system still exists (presumably having not been consumed by the Sun going red giant) and the barycenter is outside of Earth, it would still not be classified as a binary planet by today's naming method. All dominant bodies wobble when others orbit them. The barycenter didn't matter for Pluto / Charon. Sometimes even the solar system's barycenter is outside the Sun's corona. Apparently, many agree with you that barycenter location alone is not enough to matter. Neither does relative size to the host planet, origin, or a host of other factors. Perhaps if we one day find a system comprised of 2 planets of equal masses rotating around a barycenter at the midpoint of the distance between them, it'll warrant a new designation simply because astronomers would have to flip a coin to decide which is the planet and which is the moon.
There's so much variety out there that it's difficult to create meaningful definitions and categories - I don't envy the committees making such decisions.