particle physics - Overall symmetry of pion wave function

Mesons are bosons, therefore their wavefunction must be symmetric under particle exchange. Overall, the meson wave function ($\text{WF}$) has the following contributions:

$$\text{WF} = \lvert \text{flavor}\rangle \lvert \text{spin}\rangle \lvert \text{radial}\rangle \lvert\text{color}\rangle.$$

Mesons are a color singlet, their color wavefunction looks like:

$$\frac{1}{\sqrt{3}}(\lvert r\bar{r}\rangle + \lvert b\bar{b}\rangle + \lvert g\bar{g}\rangle)$$ which is symmetric.

For the pseudoscalar mesons the spin part is antisymmetric because the spins are a spin singlet.

The radial part is symmetric for the pseudoscalar/vector mesons, because the angular momentum $\ell$ is zero.

When it comes to the flavor part of the wavefunction for the pseudoscalar mesons, it is symmetric as well: for example

$$\pi^+: \frac{1}{\sqrt{2}}(\lvert u\bar{d}\rangle + \lvert \bar{d}u\rangle).$$

This gives us an overall antisymmetric wavefunction because of the antisymmetric spins. But the wavefunction has to be symmetric! The same goes for the vector mesons: there we just have an antisymmetric flavor part times symmetric spin. So it is again overall antisymmetric.

What's wrong with this reasoning?