Interpretation of the Uncertainty Principle $||f(x)||_2^2 \le 4\pi ||x f(x)||_2 ||\omega\hat f(\omega)||_2$?

Question

The Uncertainty Principle states that for any $f\in S(\mathbb{R})$ we have $$ ||f(x)||_2^2 \le 4\pi ||x f(x)||_2 ||\omega\hat f(\omega)||_2, $$ and supposedly this means that a function $f$ and its Fourier Transform $\hat f$ can't both be concentrated in a small region. Well lets take a function $f$ with a small support, e.g. $f(x) = \chi_{(0,1)}$. Then $||f(x)||_2^2 = 1$ and $||x f(x)||_2 = 1/\sqrt 3$. So we have the following condition on $f(\omega)$: $$ 1 \le \frac{4\pi}{\sqrt 3} ||\omega\hat f(\omega)||_2, $$ or $$ \frac{\sqrt 3}{4 \pi} \le ||\omega\hat f(\omega)||_2, $$ I don't see how this implies that $\hat f$ must have large support. It seems that the support of $\hat f$ could be arbitrarily small as long as its magnitude is large enough to satisfy the bound?