The main question is: Does the autor consider random function $f$ dependent on $\omega$?
It seems that the answer is yes, because at the page 125 (see 4.90) he consider function $u(t,W_t)=f(t,at+bW_t)$ and $a(\omega), b(\omega)$ are random functions. In case the answer is yes, is the proof correct?
No, $f$ does not depend on $\omega$. Or at least, not in itself. In the standard Ito lemma, we plug a stochastic function ($\xi$ in this case) into $f$. Hence, $f(t,\xi(t))$ certainly depends on $\omega$, but $f(t,g(t))$ does not if $g$ is a deterministic function. In general, the dependence of functions on $\omega$ is often not shown, but it should always be made clear when a function is introduced. Here, $f$ is introduced in theorem 4.4 without any dependence on $\omega$.