Recapitulation is more of a hermeneutic. Applied to the atonement it becomes a type of theory of how the mechanism works, and in that regard I would say it is essential, but not complete.
As a hermeneutic, recapitulation is all over the Bible and the Bible is impossible to understand without it. The Biblical writers are constantly telling the same stories (recapitulating the same stories) over and over again with slight modifications or inversions to make their point. The logic of recapitulation is "just as, so also."
For example, when Exodus says that Moses was put in an "ark" and put into the waters of the Nile, clearly the writer wants us to think of Moses as the new Noah. Just as Noah, so also Moses. Then when Joshua parts the Jordan and the Israelites pass through, clearly the writer wants us to think of Joshua as the new Moses. This type of stuff is all over the place in the Bible. Literally thousands of connections like this. Just as Moses, so also Joshua.
Again regarding Exodus, there are 7 women in the first two chapters who save the world. They are all women who "undo" the failings of Eve, and who prepare the salvation story of the Exodus through the Red Sea. Mary is yet another recapitulation of women such as these. Irenaeus is correct to say that Mary succeeds where Eve fails, and to draw all the appropriate connections. Where he oversteps is calling her an "intercessor" or advocate.