> From what I've learned so far, the set theory suffers from Russel's Paradox
So-called "naive set theory" does, but my understanding is that this paradox (and others like it) sparked a crisis in mathematics that led to the creation of ZFC and others. ZFC is not known to be inconsistent (due to deep and technical reasons, I can't claim positively that it is consistent), and it was designed to avoid the paradoxes that plagued naive set theory.
Russell's type theory was another early contender for a formalism. For whatever reason (I'm not aware of all the details), Zermelo-Fraenkel set theory (later extended with the Axiom of Choice) won the popular mindshare. Personally, I'm more a fan of the Elementary Theory of the Category of Sets (ETCS) [1], which is indeed drawn from category theory. But it's equivalent in deductive power to ZFC, so what you pick mostly only matters if you're doing reverse mathematics. (This is what really answers your question, I think.)
Russell's type theory and modern type theories are distinct (and there is no single "type theory"). I'm led to believe the commonality is primarily with the usage of a hierarchy of universes, so that entities in one unverse can only refer to entities in a lower universe.
[1] "Rethinking set theory", Tom Leinster: https://arxiv.org/abs/1212.6543