I've been watching Frederic Schuller's course on "The Mathematics and Physics of Gravity and Light" and at a moment during Lecture 6 he claims there is no basis for $\Gamma(TS^2)$ (the space of smooth sections $\sigma \colon S^2 \to TS^2$) when considered as a $C^\infty(S^2)$-module and gives an example through the claim that since every vector field must vanish somewhere, you can't get a basis. During his example, he uses a vector field which vanishes at points $(\pm 1, 0, 0)$ and $(0, 0, \pm 1)$ and says that since on, say, $(0, 0, \pm 1)$ you have only one nonvanishing vector field, and thus a linear combination can't point towards a different direction.
My first question is: adding a new vector field which vanishes at $(0, \pm 1, 0)$ wouldn't solve the problem? At every point you have at least two non-parallel vectors, so it seems to me it would work, at least in principle. Furthermore, every vector space admits a basis, so $\Gamma(TM)$ (for some smooth manifold $M$) when considered as a real vector space has a basis. Since every real number can be regarded as a constant (and hence smooth) function, wouldn't it allow us to obtain a basis for $\Gamma(TM)$ when considered as a $C^\infty(M)$-module?
I believe it is worth mentioning I do not have much background in Differential Geometry nor Module Theory.