Why is the concept of "mass extinction" a relatively recent development in geology?
The idea of mass extinction is not that recent actually: Cuvier (1798), Buckland (1823) and d'Orbigny (1851) for instance were already talking about global catastrophes in earth history, linked to extinctions. But during the same period, Brocchi (1814) and Lyell (1832) proposed that extinctions of species occurred individually and were a gradual process (either only linked to an intrinsic taxa longevity for Brocchi, or variations in the environment for Lyell). Darwin, following Lyell, also thought that extinctions were gradual and not catastrophic. He also noted the fact that hiatuses in the fossil record or artificial concentration in some strata could show apparent extinction event.

The issue with mass extinction is that to demonstrate their existence you need to be able to demonstrate extinction synchronicity and quantify the amount of species going extinct (to show that it is more than just background noise).

Demonstrating the synchronicity of one mass extinction is what Alvarez et al. 1980 managed to do thanks to the Iridium layer at the K/Pg boundary. More generally, the possibility of correlating extinctions precisely is something that evolved in par with the evolution of stratigraphic tools, and the 1970-1980s is the period during which high-resolution stratigraphic methods arose (chronostratigraphy, magnetostratigraphy, stable isotope stratigraphy for instance).

Quantifying mass extinction is what Jack Sepkoski did with his compendium of marine invertebrates (see Sepkoski 1978, 1979; Raup &amp; Sepkoski 1982, etc.). Today, the PbDb (PaleoBiology DataBase) is the project which focusses on that specific issue (see for instance Alroy et al. 2001). It still remains today the main hurdle in studying mass extinctions. 

Alroy, J. et al., 2001. Effects of sampling standardization on estimates of Phanerozoic marine diversification. PNAS, 98(11): 6261-6266.
Alvarez, L. W. et al., 1980. Extraterrestrial Cause for the Cretaceous-Tertiary Extinction. Science, 208: 1095-1108.
Brocchi, 1814. Conchiologia fossile subapennina, con osservazioni geologische sugli Apennini e sul suolo adiacente. Stamperia reale. (link to 1843 reedition)
Buckland, 1823. Reliquiae Diluvianae or, Observations on the Organic Remains attesting the Action of a Universal Deluge. Murray, London.
Cuvier, G., 1798. Mémoire sur les espèces d'éléphans vivants et fossiles. Mémoires de l'institut national des sciences et des lettres, 2: 1-32.
d'Orbigny, A. 1851. Cours élémentaires de paléontologie et de géologie stratigraphique. Masson, Paris.
Lyell, C. 1832. Principles of Geology. Murray, London.
Raup D. M. &amp; Sepkoski J. J. Jr, 1982. Mass Extinctions in the Marine Fossil Record. Science, 215: 1501-1503.
Sepkoski J. J. Jr, 1978. A kinetic model of Phanerozoic taxonomic diversity I. Analysis of marine orders. Paleobiology, 4: 223-251.
Sepkoski J. J. Jr, 1979. A kinetic model of Phanerozoic taxonomic diversity II. Early Phanerozoic families and multiple equilibria. Paleobiology, 5: 223-251.