Well, they did.

The Russians did open up a front in the Caucasus in November 1914 and this region did become the site of pretty nasty back and forth fighting between the Ottomans and the Russians. The lateness of Ottoman entry into the war prevented either side from rushing to seize an advantage in the Caucasus, and weather conditions in this mountainous region were not suited for operations in late autumn and early winter. Neither side was able to gain a decisive advantage over the other during the conflict. Even though the Russian Army was much larger than the Ottoman forces, the bulk of its troops were needed on the Eastern Front, not in the Caucasus. Terrain also gave the defenders an advantage as logistics and other sundry details were much harder in mountainous country.

The biggest impact the Caucasus campaign had was arguably its impact on the local population, especially the ethnic Armenians in the theater. The war engendered long-standing stereotypes within the Ottoman government and military about the untrustworthiness of this population. The Russians for their part did not discourage these sentiments and tried to appeal to Armenians and other non-Muslim Anatolian minorities to rise up against the Ottomans. These Russian entreaties were largely ineffective but only provided further fuel for Ottoman ethnic cleansing and the Armenian genocide.

Sources

Bartov, Omer, and Eric D. Weitz. Shatterzone of Empires: Coexistence and Violence in the German, Habsburg, Russian, and Ottoman Borderlands. Bloomington, Ind: Indiana University Press, 2013.

Reynolds, Michael A. Shattering Empires The Clash and Collapse of the Ottoman and Russian Empires, 1908-1918. Cambridge [u.a.]: Cambridge Univ. Press, 2012.