Why do physicists use wave functions with more than two components?

For $n$-body systems you just need a single component wavefunction.
For example, for a two-body system you would need a wavefunction of 6 variables.

$psi(x_1,x_2,y_1,y_2,z_1,z_2)$

That satisfies the equations

$ihbarfrac{partial psi}{partial t}=-frac{hbar^2}{2m}(frac{partial^2 psi}{partial x_1 ^2}+frac{partial^2 psi}{partial y_1 ^2}+frac{partial^2 psi}{partial z_1 ^2})+V(x_1,x_2,y_1,y_2,z_1,z_2)psi$

$ihbarfrac{partial psi}{partial t}=-frac{hbar^2}{2m}(frac{partial^2 psi}{partial x_2 ^2}+frac{partial^2 psi}{partial y_2 ^2}+frac{partial^2 psi}{partial z_2 ^2})+V(x_1,x_2,y_1,y_2,z_1,z_2)psi$

In the case of fermions you just need a wavefunction with two components. For the Dirac equation it becomes a spinor field with four components. In what case would you need more components?