Efficiently finding "unfounded edges" in a strongly connected graph
I believe that computing graph dominators will solve your problem. This can be done in $O(|E|\log |V|)$ time (or even faster, though they describe the faster algorithm as "sophisticated") using a 1979 algorithm by Lengauer and Tarjan, which is (at least currently) described here (21 pages -- I have only read the first few pages myself).

Add a new "start" node with an out-edge to every source node, run the dominator algorithm to determine dominators for every node, and then check each edge $uv$: $uv$ is unfounded if and only if $v$ dominates $u$. (Each of these checks can be done in constant time using a lowest common ancestor (LCA) algorithm, although that may be overkill here -- for an edge $uv$, we only care about whether or not $LCA(u, v) = v$.)