As it was put to me, one major difficulty with the breeding capability definition of species is for Palaeontologists dealing with ancient species. Clearly, they cannot apply such a test and thus have to use different criteria. For modern species, breeding capability is the usual definition, but it is a much misunderstood point. I have often seen people talk about breeds of dogs, and the impracticality of a very large dog cross-breeding with a toy breed. But that is not the point. It is all a simple question of whether a gamete of one sex from one organism can fuse with a gamete of the other sex from another organism to produce a viable zygote. That is a matter of how well the chromosomes match up, and is not necessarily indicated by obvious physical differences in the parent organisms, however extreme.
There is also, of course a further subtlety around so-called ‘hybrid’ species. Famously it is possible to cross a lion and a tiger, which certainly does prove that, in evolutionary terms, their divergence was relatively recent. However, the Liger is necessarily infertile – its gametes will fuse with nothing, not even that of another Liger. Likewise for mules and various other hybrid species. So the key for definition as the same species is that gametes from each can produce a viable zygote that can develop into a fertile adult. That definition works pretty well for animal species, but even that is much more open to being undermined by the realities of what is possible with the intervention of human kind when it comes to plant species.