ag.algebraic geometry - the proof of "theorem of connectedness"

A typical context in which one has the condition $f_{*}mathcal O_X = mathcal O_Y$
in that in which $f$ is a birational morphism and $Y$ is normal. In this context, the proof
that the fibres are connected is due to Zariski (I believe that it's the original version
of his ``main theorem'') and certainly predates EGA methods. One can find the paper in his
collected works. (It's been a long time since I looked at it, but I would guess that his main
technical tool is valuation theory; I might check this when I get a chance.)

However, it is worth bearing in mind the evolution of Zariski's work on this kind of question:
his investiations of this sort of connectedness theorem culminated in his proof of his connectedness theorem, to the effect that a specialization of connected varieties is again
connected. His was the first purely algebraic proof of this result, I think. To give the proof, he invented his theory of formally holomorphic functions, which I believe was regarded
at the time as being the most difficult part of the algebraic theory of algebraic geometry
developed by Weil, Chevalley, and Zariski. This theory served as one of the inspirations for the theory of formal schemes, and is the precursor to Grothendieck's theorem on formal functions (the proof via formal schemes mentioned in the question).

If so great a geometer as Zariski was led to introduce these kinds of formal methods to study connectedness problems, it is probably reasonable to regard them as somewhat intrinsic to the problem.