A noetherian proof of Zariski's Main Theorem?

I would suggest chapter IV of the 1970 book "Anneaux Locaux Henséliens", by Michel Raynaud published in Springer Lecture Notes in Math no. 169. It gives a very general proof, way simpler than the one in EGA IV and, in my opinion, very readable. The proof is based in a paper by Peskine from 1966. The proof in Raynaud's book is complete, as far as I can recall.

As a footnote, sometimes noetherian hypothesis do not make arguments simpler, but, of course, this depends on the issue at hand.

There's a purely algebraic proof in some lecture notes by Mel Hochster. He explains the translation into the language of varieties, as well.

Raynaud and Hochster, and Stacks, give essentially the proof of Peskine this proof does not use noetherianity

a constructive proof, extracted from Peskine proof is given in the following paper

{Alonso, M. E. and Coquand, T. and Lombardi, H.}, TITLE = {Revisiting {Z}ariski main theorem from a constructive point of view}, FJOURNAL = {Journal of Algebra}, VOLUME = {406}, YEAR = {2014}, PAGES = {46--68},