The proofs I have seen of the Schur property on $\ell_1$ use the gliding hump, or Vitali-Hahn-Saks which in turn, is proved using Baire. I am wondering if it can be done directly with Baire, but without the measure theory.
I think the answer is yes but my proof needs work. The key ingredient is the fact that the closed unit ball $\overline B$ with the weak$^*$ toplogy, of $(\ell_1)^*$, is metrizable. For then if we define $A_n=\{f\in (\ell_1)^*:f(x_n)\le\epsilon\}$, where $x_n\rightharpoonup 0$, then it's not hard to see that the $\bigcap_{n=k}A_n$ are weak$^*$ closed and $\bigcup_k\bigcap_{n=k}A_n=(\ell_1)^*$, so for some integer $k,\ \bigcap_{n=k}A_n$ contains a ball $B'\subseteq \overline B$ (of course, in the subspace topology on $\overline B$).
Assuming that this ball can be taken to be a weak$^*$ neighborhood of $0$, we then find a $y\in\ell_1$ such that the subbasis element $V_y=\{f\in (\ell_1)^*:|f(y)|<1\}$ satisfies $V_y\cap\overline B\subseteq \bigcap_{n=k}A_n.$
Choose $N$ such that $\sum_{N+1}|y_i|<1.$
Of course, $(\ell_1)^*=\ell_{\infty}$. Using this fact, define \begin{equation*} z_n = \begin{cases} \frac{x^n_i}{|x^n_i|} & \text{if } i>N\\ 0 & \text{otherwise}\\ \end{cases} \end{equation*}
and note that $\sum_i |y_iz_n^i|=\sum_{i=N+1} |y_i|<1$, so $z_n\in V_y$. It follows that
$\|x_n\|=\sum_{i=1}^\infty |x_n^i|=\sum_{i=1}^N |x_n^i|+\sum_{i=N+1}^\infty |x_n^i|=$
$\sum_{i=1}^N |x_n^i|+\sum_{i=1}^\infty |z_n^ix_n^i|.$
But $z_n\in V_y\cap \overline B\subseteq \bigcap_{m=k}A_m$ so the second term is less than $\epsilon$ as soon as $n>k.$ To finish, let $n\to \infty$ in the finite sum and note that the projections are weak$^*$ continuous.
I think this proof is substantially correct, but there are a few sticking points. I am interested in getting it right, and would like to know if there are any generalizations of this idea/technique, assuming of course, that it works.