Your condition that $\| S\|<\sin\frac{\pi}{2n}$ implies that the angle between $u_{k+1}$ and $u_k$ is less than $\pi/2n$, for $k=0,...n-1$. Therefore the angle between $u=u_0$ and $u_k$ is $<\pi$ for $k=1,...,n$. Since $u\in L$, $u_1,...,u_n$ cannot be orthogonal to $L$ that is $Pu_k\neq 0$.

Your condition that $\| S\|<\sin\frac{\pi}{2n}$ implies that the angle between $u_{{k+1}}$ and $u_{k}$ is less than $\pi/2n$, for $k=0,...,n-1$. Therefore the angle between $u=u_0$ and $u_{k}$ is $<\pi/2$ for $k=1,...,n$. Since $u\in L$, $u_1,...,u_n$ cannot be orthogonal to $L$ that is $Pu_k\neq 0$.