= Solution
Assume $V_0\ne0$. <Linear independence> of the <Fourier sine basis> makes a flat string equivalent to vanishing of all three temporal coefficients. The fundamental requires $\sin(\omega_1t)=0$, which automatically zeros the other two. Thus the earliest positive return is
$$
\boxed{t_{\rm flat}=\frac{\pi}{\omega_1}=\frac Lc.}
$$
The <velocity> need not have returned to its initial value at that time; full recurrence of the displacement-and-velocity data occurs at $2L/c$.
A nonzero pure fundamental profile would require $\sin2\omega_1t=\sin3\omega_1t=0$. The common zeros have $\omega_1t$ an integer multiple of $\pi$, so the fundamental vanishes too. A pure second or third harmonic would already require the fundamental to vanish, again making the whole string flat. \b[The string never has a nonzero pure sine profile.] If $V_0=0$, it remains identically zero and there is no earliest positive isolated return.
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