I am wondering if my following reasoning is correct.
I have a series $\sum_{i=1}^\infty X_i$ that converges in $L^p(\Omega)$ to some $X$. $L^p$ convergence implies a.e. convergence of a subsequence, hence $\sum_{i=1}^{n_k}X_i$ converges almost everywhere to $X$ for some increasing $(n_k)_k$. But by definition the limit of this subsequence is the same as $\sum_{i=1}^\infty X_i$, so this series also converges a.e. to $X$.
The limit in $L^p$ will coincide with the almost-everywhere-limit of the subsequence. However, this does not imply that the whole sequence converges almost everywhere.
Classical counterexample: Write $Y_k:=\sum_{i=1}^k X_i$. Let $\{Y_k\}$ be the sequence of indicator functions $$\chi_{[0,1/2]},\chi_{[1/2,1]},\chi_{[0,1/4]},\chi_{[1/4,1/2]},\dots$$ This sequence converges in $L^p([0,1])$ to $0$. It has an almost everywhere convergent subsequence: $$\chi_{[0,1/2]},\chi_{[0,1/4]},\chi_{[0,1/8]},\dots$$ Yet, it converges nowhere.