We talk about science. The art of science is it to put the real world into categories. No categories no science.

I wonder what "Hybridization is very common in the animal kingdom" does mean. Any data? Usually Hybrids are sterile, so there is no effect on natural occurring species - see here. And: Hybrids are no species. I understood that occuring hybridization is a result of the anthropocene.

Taxonomy may be old but species is the central unit of evolutionary biology. The concept was developed by Ernst Mayr, who knew about DNA. Indeed the idea behind it all is that genes can be shared by all members of a given species. When gene flow is hampered by barriers populations can differentiate. Genetic differentiation will lead to phenotypic differentiation. New species are born.

It is only flimsy if used incompletely: Two organisms belong to two different species if they do not reproduce - the keyword would be „reproductive community“. This can be due to genetical /anatomical differences or (!) because they don’t share the same ecological niche.

This definition does not apply to organisms that reproduce non-sexually.