Why Doesn't Compressive Sensing Work for Any Signal?
You can measure and reconstruct such a 1-sparse signal as you describe in your question. The crucial misunderstanding here is, as @MBaz points out, that you have to know the basis $\Psi$.

You can take measurements in a universal manner without knowing the dictionary $\Psi$ beforehand, if you choose the measurement matrix $\Phi$ as a sufficiently random matrix - could be a random Gaussian matrix. What universality gives you is the ability to reconstruct $x$ again with good accuracy almost no matter what $\Psi$ turns out to be. You can take such universal measurements at the encoder without knowing $\Psi$, but you still have to know some $\Psi$, that admits a sparse representation of $x$, (and $\Phi$) at the decoder.