Prove or disprove: For every vertex v in a graph G without isolated vertices, there is some maximum matching that saturates v.
I think this is true: Suppose that vertex $v$ is saturated by some matching $M$. If $M$ is maximum we are done. Otherwise, we know that there is an augmenting path that enlarges $M$. Although we may lose edges in the process of using augmenting paths, the vertices saturated by $M$ remain the same. So $M$ will now be a maximum matching saturating $v$. Is it correct?
I am convinced. You should however start with a matching that definitely saturates $v$, which is only possible because $v$ is not isolated (this way you use the hypothesis at all) - take $M$ to be any edge incident to $v$ and then proceed to augment, as you proposed.