No.

Choose a basis $V_1, V_2, V_3$ of Killing fields

Note that
$$\int\limits_{\mathbb S^2} V_iu\cdot x_j\cdot d{\mathrm area}
=
-\int\limits_{\mathbb S^2} u\cdot V_ix_j\cdot d{\mathrm area}$$
Threfore is sufficient to find a fnction which is orthogonal to each of 9 functions $V_ix_j$.