The focus these days is on genetic isolation. Generally, a species would be defined as a genetically compatible interbreeding population, or group of populations, that is genetically isolated from other groups.

Reproductive compatibility is a component of that, in that successful reproduction is important for an interbreeding population to exist, but it's not the sole or primary requirement. I'm an entomologist, and I've always been a fan of the morphological species concept, but genetics has shown that multiple genetically distinct species can exist under the same morphological umbrella. Also, some species that are morphologically distinct may not be genetically isolated enough to actually be described as distinct species. The morphological species concept is still used, but not many people want to spend their lives dissecting beetle genitalia under a microscope when they can just grind up a tarsal segment for DNA barcoding instead.

Phylogenetic relationships are complex enough that they can be confusing for high school and college students, as well as adults without a strong biology background. Couple that with the fairly terrible science curricula in the US, and you end up with broad acceptance of the biological species concept.