Suppose we have a continuous local martingale $M$. Then, by definition, there exists a sequence of stopping time $(T_n)$ s.t $T_n\uparrow \infty$ and that the stopped process $M^{T_n}$ is a uniformly integrable martingale
Now suppose we fix $s >0$. Then if $M$ is a continuous local martingale then there will exist some $n$, such that $T_n>s $ a.s.
Since the stopped process $N_t=M_{\min\{t, T_n\}}$ is a martingale, then we can stop it once again to get that $P_t=N_{\min\{t,s\}}$ is a martingale.
But $P_t=M_{\min\{s,t,T_n\}}=M_{\min\{s,t\}}$ as we have $T_n>s$ a.s.
But that means that for a fixed time $s$ stopping the continuous local martingale will result in a martingale.
Similarly, this idea can be generalised to the case when $s$ is bounded.
Is all my reasoning correct? It seems not, because when reading proofs of various statements I came across one which says "Stopped continuous local martingales at fixes times are continuous local martingales" so I'm starting to doubt my reasoning