The flaw is in the clause such that every set contains at least one point that no other does. The ordinal space $\omega_1$ with the order topology is sequentially compact but not compact. The most obvious open cover with no finite subcover is the one consisting of the sets $V_\alpha=\{\xi:\xi<\alpha\}$ for $\alpha<\omega_1$: this is an uncountable nest of increasing open sets, so each point of the space is actually contained in uncountably many of the $V_\alpha$.
Theorem: $X$ is countably compact, then $X$ is strongly limit compact: every countably infinite subset $A$ has an $\omega$-limit point, i.e. a point $x$ such that for every neighbourhood $U$ of $x$ we have $U \cap A$ is infinite.
Proof: suppose not, then every $x \in X$ has a neighbourhood $O_x$ such that $O_x \cap A$ is finite. Now a nice trick: for each finite subset $F$ of $A$ (and there are countably many finite subsets of $A$) define
$$O(F) = \bigcup\{O_x: O_x \cap A = F\}$$
As every $O_x$ is a subset of one of the $O(F)$ (namely that with $F = O_x \cap A$), and the $O_x$ cover $X$, the $O(F)$ form a countable cover for $X$.
Hence there is a finite subcover $O(F_1), \ldots, O(F_N)$ but then there is some $a_0 \in A \setminus \bigcup_{i=1}^N F_i$ (as the $F_i$ are finite subsets of $A$) and this $a_0$ is not covered by any of the $O(F_i)$ for $i \le N$, and this is a contradiction. So $A$ does have an $\omega$-limit point.
BTW the reverse also holds, as you can see here, from which I also borrowed the above argument.
But now the sequential compactness can be proved: let $(x_n)$ be a sequence in $X$. If $A = \{x_n : n \in \mathbb{N} \}$ is finite, there is a constant (hence convergent) subsequence. So we can assume $A$ is infinite. So $A$ has an $\omega$-limit point $p \in X$, as we saw above.
Let $U_n, n \in \mathbb{N}$ be a countable local base at $p$. Then pick $n_1$ with $x_{n_1} \in U_1 \cap A$. Then having chosen all $x_{n_k} \in (U_1 \cap \ldots U_k) \cap A$, for some $k \ge 1$, where we also have $n_1 < n_2 <\ldots < n_k$, we note that $\cap_{i=1}^{k+1} U_i$ is an open neighbourhood of $p$, so contains infinitely many points of $A$, so in particular we can pick $n_{k+1} > n_k$ such that $x_{n_{k+1}} \in \cap_{i=1}^{k+1} U_i$. Continue this recursion.
It's now standard to check that $x_{n_k} \to p$ is a convergent subsequence of $(x_n)$.
Because I used $\omega$-limit points, there was no need for $T_1$-ness (which you can have with just limit points).
Best Answer
The finite case is trivial. Consider {N(x_1),N(x_2),N(x_3),...} an arbitrary countable open cover. {x_1,x_2,x_3,...} partitions into finitely many subse uences all of which are convergent. Without loss of generality {x_1,x_2,x_3,...} converges to x_infinity. But x_infinity is in N(x_k) for some k. It follows that we have a finite subcover.