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Given an SDE for an underlying:

$$dS(t) = mu(S,t)dt+sigma(S,t)dW(t)$$

the SDE for the value of the option $V=V(S,t)$ is given via Ito's lemma as:

$$dV = V_tdt+V_Smu(S,t)dt+frac12V_SSsigma^2(S,t)dt+V_Ssigma(S,t)dW(t)$$

It seems that this would results in an SDE containing $S(t)$.

How does one then obtain an SDE for the option value so that it can be simulated directly without simulating the underlyings, i.e. something like

$$dV(t) = m(V,t)dt+s(V,t)dW(t)?$$

That's impossible. The value of your option is a function of time and the value of the underlying:
$$ V(t) triangleq V(t,S_t) $$
How can you possibly know the value of the option at time $t$ without knowing the value of its underlying? Equivalently, you need to know how your underlying has evolved in order to know how the value of your option has evolved. You will always have a dependency between the value of an option and its evolution on the one hand, and the value of the underlying on the other hand.

This has been indirectly discussed in this question. We assume that $mathcalF_t, , tge 0$ is the natural filtration generated by the Brownian motion $W_t,, t ge 0$. Moreover, let
$B_t= e^int_0^t r_sds$ be the money market account value at time $tge0$. Let $V_T$ be the option payoff at maturity $T$. Then
beginalign*
V_t = B_tE_Qleft(fracV_TB_Tmid mathcalF_t right),
endalign*
where $E_Q$ is the expectation operator under the risk-neutral probability measure $Q$. Note that, $V_t/B_t, t ge 0$ is a martingale. Therefore, by the martingale representation theorem, there exists an adapted process $kappa_t, 0le t le T$ such that
beginalign*
fracV_tB_t &= V_0 + int_0^t kappa_u dW_u.
endalign*
Then
beginalign*
dV_t &= dleft(B_t fracV_tB_t right)\
&=r_tV_t dt + B_t kappa_t dW_t\
&=V_tBig(r_t dt + s(V,t) dW_tBig).
endalign*
where $s(V,t)=fracB_t kappa_tV_t$.