@naturallyInconsistent sorry for pinging you, if that bothers you I won't do it again. Did you see my reply above?
What I was trying to say is that unlike the example I provided with $A(t_3)$, the $U(\beta,0)$ outside $T$ in $(10.15)$ and the one inside in $(10.16)$ are not the same. The latter comes from $TU(\beta,\tau)U(\tau,\tau')U(\tau',0)$. That's what I meant when I said "we don't just move it inside" (unlike my $A(t_3)$)