Understanding characterizations of Matching on Graphs
It is nice for OP to extract the essence of the proof for Berge's Lemma, which is described as the  "Theorem (I think it is due to Peterson)" to create the hypothesis.

"Is the hypothesis correct? If not, why is it wrong?" Yes, it is correct.

"What about the hypothesis with respect to non-bipartite graphs?" Yes, it is still correct with respect to non-bipartite graphs.

"If the hypothesis is correct, why is always Egervary's Theorem used?" Well, I cannot avow Egervary's Theorem is always used (when? and where?).
It is true, however, that in the case of bipartite graph, Egervary's Theorem is instrumental in creating a fast and easily-implementable algorithm to find the maximum weight matching. In comparison, I cannot find  a similarly good algorithm for which the hypothesis can be equally helpful. So it is no wonder that Egervary's Theorem is usually mentioned when we come to the maximum weight matching for bipartite graph.