Can Tarjan's SCC algorithm find satisfying assignment to just 'any' digraph?
You have a faulty premise.  There is a 2SAT instance corresponding to $(a \implies b) \land (b \implies \neg c)$.  Just note that $x \implies y$ is equivalent to $\neg x \lor y$.  Then you can transform the original formula to $(\neg  a \lor b) \land (\neg b \lor \neg c)$, which is obviously a 2SAT instance.

That algorithm you refer to can solve 2SAT, but it can't solve anything stronger as is.  However, there are related graph-based algorithms to test satisfiability of arbitrary conjunctions of Horn clauses.