Is $0$ the only vector in the kernel of every bounded linear functional?

Take a subspace $L=span\{x\}$ and then define $f:L\to K$ by $f(\lambda x)=\lambda$. This is clearly a bounded linear functional such that $f(x)\ne 0$. By Hahn-Banach theorem you can extend it to a bounded linear functional on $X$.

Use Hahn Banach to extend any non zero linear function defined on the vector space gnerated by $x_0$ to $X$.