Marshall Hall's The Theory of finite groups only cites an asymptotic bound: a permutation group of degree n that isn't Sn or An can be at most t-transitive for t less than 3 log n. I suppose that was the state of the art at the time (late 1960s). There is an earlier paper by G.A. Miller on JSTOR that you can find by searching for "multiply transitive group".
There is a classical theorem of Jordan that classified sharply quadruply transitive permutation groups, i.e., those for which only the identity stabilizes a given set of 4 elements (from Wikipedia).
As Andy says, people didn't know until fairly late in the classification whether there were finitely or infinitely many sporadic finite simple groups. Actually "sporadic" has a fairly specific operational meaning; it means finite simple groups that are not prime cyclic, alternating, or Chevalley type. (The finite simple groups of Chevalley type are basically Lie groups over finite fields, with the twist that there are some extra ones in characterestic 2 and 3. The Tits group is usually counted with these even though it's not part of an infinite sequence. Arguably a prime cyclic group is also a Lie group over a finite field.)
Another example is the theorem on highly transitive permutation groups. The classification implies that there are no 6-transitive permutation groups on $n$ points other than $A_n$ and $S_n$. I have heard that without the classification, there is no bound which is uniform in $n$.
People certainly think that it is a good project to improve the classification of finite simple groups in general. In fact Gorenstein's announcement that the classification was complete was controversial, because there was clearly interesting mathematics left to be discovered even though there was sort-of enough at that time to believe the classification. But you have to study the classification to know what needs to be improved. If people do not have an a priori argument that there are only finitely many sporadic groups, then it would be great to have one, but I don't see how you can know in advance that you should look for that.
Note that in the modern classifications of complex simple Lie algebras or compact simple Lie groups, you do not prove that there are only finitely many exceptional ones before proving the full classification using root systems and Dynkin diagrams. On the contrary, both in the classification and in the representation theory that comes after, people are the happiest when they can treat all of the root systems uniformly, i.e., when the exceptional Lie algebras are not treated as exceptions. The fact that sporadic groups are more exceptional than exceptional Lie algebras could be one reason that the current classification of finite simple groups seems unsatisfying.
Best Answer
Nikolov and Segal proved in On finitely generated profinite groups II, products in quasisimple groups that finite-index subgroups of finitely generated profinite groups are open. This implies that the topology in such a group is uniquely determined by the group structure. They use the classification in a crucial way.