Even today, we don’t have a definition of “species”, a definition acceptable to biologists working in this domain. What is needed is a theory, and further development of that theory.
Those who quibble about definitions should develop theories and test them, amend them, replace them. All these activities help understand the phenomenon in the question. When people engage in definitional disputes, it shows that intuitions about the phenomenon conflict. That’s all. To settle these issues, one should do science: that is, develop theories, derive consequences, add back ground theories to test these consequences.
Incidentally, the species debate is fairly simply summarized / resolved, at least for typical sexual organisms: There is nothing whatsoever in first-principles evolutionary biology that says that these can always be partitioned into discrete categories that one might call "species". What evolutionary biology says is
1. When two populations overlap in space and time and have the opportunity to interbreed, they might do so.
2. In the absence of interbreeding, populations evolve, divergently w.r.t. most markers other than the occasional selection-mediated convergence.
So, divergence might be the result of lack of opportunity to interbreed (this allows genetic and phenotypic diversity to accrue, but does not itself represent a biologically real division), or it might be the result of inability to interbreed (biologically meaningful, but impossible to discern if opportunity to interbreed is absent). And that's it, for sexual organisms that is the gist. In general there's nothing to say that our systems of related populations spread through space and time can always be divided into "species". On the other hand any cursory look around a single locality shows that many discrete biological entities do exist; there's nothing wrong with giving them the name "species" as long as we acknowledge that for closely related allopatric populations the question of whether they are the same "species" is not defined.
There are various problems. One is that some humans are attached to the idea that entities exist for which the word "species" is apt; or put another way they are convinced that "species" must mean _something_ and they ask what (e.g. because they like their neatly divided up bird lists).
Another is that if scientists go around saying "animals can't always be divided into things which it makes sense to call 'species'" then it significantly weakens certain arguments for the protection of populations in conservation biology / legal contexts, since these invariably name the species-level taxa they are intended to protect. Since the modern trend is towards splitting subspecies (i.e. geographical races) into species (as opposed to lumping), many conservation initiatives are based on protecting "species" for which the species designation is subjective. That doesn't mean they are flawed of course -- they are nearly always protecting valuable ecosystems. It's just that as we split geographical races, we (by definition) create rarer and more vulnerable taxa, and so conservation initiatives arise around these.
Those who quibble about definitions should develop theories and test them, amend them, replace them. All these activities help understand the phenomenon in the question. When people engage in definitional disputes, it shows that intuitions about the phenomenon conflict. That’s all. To settle these issues, one should do science: that is, develop theories, derive consequences, add back ground theories to test these consequences.