The Moon doesn't actually orbit the Earth. Or it does, but only in a geocentric frame of reference. From the heliocentric frame of reference, the moon orbits the sun in a circle, except that this circle is pushed out into points here and there. More technically, the moon's orbit is broadly a triskaidecagon - a 13-sided polygon - with very rounded angles: the gravity of the earth, trying to occupy the same orbit as the moon, sometimes pushes the moon toward the sun for a bit (the flat sides of the polygon) and sometimes pushes it out away from the sun for a bit (the rounded angles of the polygon). The moon never reverses its direction around the sun, and the moon never loops. The moon's orbit is never concave - that is, the moon's orbit never curves away from the sun, but always curves toward it.
This is an incredibly naive interpretation. A similar argument could be given for any object that follows a 'roughly elliptical' path around the sun. In other words, you could make an equal argument that the ISS orbits around the sun, or that Saturn's rings orbit around the sun. This interpretation isn't strictly wrong if you use a silly definition, but when you use a silly definition it is not true. Simply put, if the moon were orbiting around the sun then it would have a smooth orbit that appears strictly like a conic section
. That is, it would be elliptical or parabolic. Locally, the moon is dominated by the Earth's gravitational field and thus orbits the Earth, not the Sun.
This is just assertion! No true scotsman orbits the sun without a perfectly smooth elliptical or parabolic orbit. So, for instance, Pluto doesn't orbit the sun either? Your link asserts, "If the Moon wasn’t orbiting Earth, it would go into a smooth orbit around the Sun, and have no regular gravitational interaction with the Earth" - but if having regular gravitational interaction with the earth is proof of orbiting, then the earth has dozens of moons. Near earth asteroids have regular gravitational interaction with the earth. The orbit of asteroids like Cruithne is affected by the gravity of the earth just as the moon is - and, like the moon, these asteroids orbit the earth in a geocentric frame of reference (but not in a heliocentric frame of reference). So why do you include the Moon but not the other asteroids?
Some other interesting points from your link - "the case is pretty compelling" (vs your "its incredibly naive"), "because the Moon actually orbits around the Earth; it’s orbiting the Earth" [punctuation sic]. And although you define it as "gravitationally dominated" by the earth, your link links to a page that concedes that "the Moon's orbit is primarily determined by the gravitation pull from the Sun".
Convex orbit? Check. Primarily determined by the pull from the sun? Check. Closed loop about the earth? Not check. Closed loop about the sun? Check. Would removing the earth significantly change the motion of the moon? No, not really.
The only counterargument is, as he says, that the moon orbits the earth, and that therefore the moon orbits the earth. Rephrasing this as "is within the hill sphere of the earth" and defining the hill sphere as the sphere inside which objects orbit the earth is just playing with semantics - the only definition they're realy using of orbit is "is said to orbit".
Qi is not a science show. It's a trivia show that aims to be funny, not true. Occasionally they're right, but take every single thing said with more than a few grains of salt.
I don't follow. I was disagreeing with QI, which said that Cruithne was a moon of earth. And you'll note that i've just criticised QI on another page. The moon orbiting the sun has nothing to do with QI. Please read with more attention to what is actually said.
But the river tripped on her by and by, lapping
as though her heart was brook: Why, why, why! Weh, O weh
I'se so silly to be flowing but I no canna stay!