Discordance has always been a problem with species concepts, a problem that has become increasingly acute given modern genomic data. Ecological criteria often do not pick out the same entities as breeding criteria. Phylogenetic criteria often do not pick out the same entities as either ecological criteria or breeding criteria. Phylogenies of different genes often pick out different entities. This obvious discordance has prompted some (including the present author in an earlier incarnation) to advocate pluralistic species concepts.
What we were all missing is something only Darwin has seen clearly: there is no real species level. Divergence happens, driven by various factors external and internal to the organism. At some point we might call something a species for the sake of convenience, but nothing magical has happened at that point. Lineages (a diachronic connection between ancestor and descendant) and clades (a synchronic set of relatives forming a monophyletic group) are real, but not any particular taxonomic rank, including species.
Likewise, the traditional cladistic concept of monophyly has become problematic given the discordance revealed by modern genomic data. Given problems of lineage sorting, and horizontal transfer that may happen at any depth in the tree of life, we can no longer require monophyly of a set of organisms to be supported by all genes in the genome (the co-called exclusivity criterion). There are no monophyletic groups, at any level, using such a strict definition. Instead, monophyly needs to be viewed as an ensemble characteristic of descent in a set of organisms -- a monophyletic group is one supported by a majority of gene phylogenies, with incongruent gene phylogenies needing a separate explanation.
Discordance is telling us that we need to get rid of the species rank entirely and concentrate on finding, naming, and studying monophyletic groups at all levels using a rankless classification system like a suitably modified PhyloCode. The species rank is a long-outdated remnant of pre-Darwinian, creationistic biology. It turns out that the reason it has been impossible over the last century to define the species rank coherently is because it corresponds to nothing real. It's time for biologists to move on, and finally catch up with Darwin!