I'm going over Professor Tao's presentation of the Birkhoff-Kakutani theorem and I don't see how it follows that $j = (g \mapsto \tau_gf)$ (between "Lemma 2" and "Remark 2") is continuous.
I don't even see how you'd apply that uniform continuity property of $f$ because to show continuity of $j$ at $x \in G$ requires that for all $\epsilon>0$ we can find a neighbourhood $U$ of $x$ such that
$ \|j(x) - j(y)\|_\infty = \sup_{g \in G} |f(x^{-1}g) - f(y^{-1}g)| < \epsilon \text{ for all }y \in U.$
But the uniform continuity property of $f$ doesn't seem to apply to this at all.