Suppose $X^n_t$ is a sequence of martingales on a filtered probability space $\left(\Omega,\mathcal{F},\mathbb{P},\left(\mathcal{F_t}\right)_{t\in\left[0,T\right]}\right)$, that is for $\Delta>0$ it holds that $\mathbb{E}\left[X^n_{t+\Delta}\mid\mathcal{F}_t\right]=X^n_t$ for all $n$. Suppose that $X^n_t\stackrel{L^1}{\longrightarrow}X_t$ for all $t$, that is $$ \lim_{n\rightarrow\infty}\mathbb{E}\left[\left|X_t^n-X_t\right|\right]=0,\quad t\in\left[0,T\right]. $$
My guess is that $X_t$ is a martingale. I think that this is a well established result but I didn't find an explicit proof. Here is my attempt to prove it
\begin{eqnarray} \left|\mathbb{E}\left[X_{t+\Delta}\mid\mathcal{F}_t\right]-X_t\right| & = &\left|\mathbb{E}\left[X_{t+\Delta}-X_t\mid\mathcal{F}_t\right]\right|\\ & \leq & \mathbb{E}\left[\left|X_{t+\Delta}-X_t\right|\mid\mathcal{F}_t\right]\\ &=& \mathbb{E}\left[\left|X_{t+\Delta}-X_{t+\Delta}^n+X_{t+\Delta}^n-X_t\right|\mid\mathcal{F}_t\right]\\ &\leq & \underbrace{\mathbb{E}\left[\left|X_{t+\Delta}-X_{t+\Delta}^n\right| \mid\mathcal{F}_t\right]}_{A^n}+\underbrace{\mathbb{E}\left[\left|X^n_{t+\Delta}-X_{t}\right| \mid\mathcal{F}_t\right]}_{B^n}. \end{eqnarray}
Now it should be that $A^n\stackrel{a.s.}{\rightarrow} 0$ since $X^n_{t+\Delta}\stackrel{L^1}{\longrightarrow}X_{t+\Delta}$, nevertheless I don't know how to use the martingale property of the $X^n$'s, I think that this should implies that $B^n\stackrel{a.s.}{\rightarrow} 0$ which would finally imply that $$ \mathbb{E}\left[X_{t+\Delta}\mid\mathcal{F}_t\right]=X_t,\quad \mathrm{a.s.}. $$
It seems that your bound are too crude. We have \begin{align} \mathbb E\left[X_{t+\Delta}\mid\mathcal F_t\right]-X_t&= \mathbb E\left[X_{t+\Delta}\mid\mathcal F_t\right]-X_{t}^{(n)}+X_{t}^{(n)}-X_t\\ &=\mathbb E\left[X_{t+\Delta}-X_{t+\Delta}^{(n)}\mid\mathcal F_t\right]+X_{t}^{(n)}-X_t&\mbox{ by the martingale property}. \end{align} Therefore, by Jensen's inequality, $$\left|\mathbb E\left[X_{t+\Delta}\mid\mathcal F_t\right]-X_t\right| \leqslant \mathbb E\left[\left|X_{t+\Delta}-X_{t+\Delta}^{(n)}\right|\mid\mathcal F_t\right]+\left|X_{t}^{(n)}-X_t\right|,$$ and taking the expectation, we get $$\mathbb E\left|\mathbb E\left[X_{t+\Delta}\mid\mathcal F_t\right]-X_t\right| \leqslant \mathbb E\left|X_{t+\Delta}-X_{t+\Delta}^{(n)}\right|+ \mathbb E\left|X_{t}^{(n)}-X_t\right|,$$ which converges to $0$ as $n$ goes to infinity.