A note on terminology: as used in modern-day historiography, "the Silk Road" does not refer to any specific trade route across Eurasia. While the term was invented in 1877 by German geographer Ferdinand von Richthofen to refer to a particular route used in Antiquity and identified by Ptolemy, the term has since expanded into a shorthand way of referring to all overland trans-Eurasian trade routes. The popular image of an actual road stretching across Eurasia, frequented by merchants traveling all the way from China to the Mediterranean, is a myth. So both the "Steppe Route" and the "Silk Road" referred to in your post would be referred to in modern historiography as part of "the Silk Road." The distinction between the two would be that between the "northern route" or the "southern route," both of which refer again not to any specific road, but to the vague, general path across which the goods traveled.

That being said, the relative significance of these varying routes changed many times in history and thus without more detail, I don't know which instance of the southern route overtaking the northern route you're referring to. Do you have a particular time period in mind?

I have heard that conflict between Muslims and Christians lead European powers to seek new trade routes to Asia. Was the Steppe Route considered?

For the Spanish and Portuguese explorers who first sought to find a way to directly access the goods of the Indies, conflict with Muslims was less significant than the fact that they were simply in the way. Western Europeans were not cut off from trade with the east by Muslim states (as I sometimes see claimed), rather the Muslims operated as middlemen. Also operating as middlemen were other Europeans - the primarily Italian (and above all Venetian) traders who dominated Mediterranean commerce. For the Iberians, situated on the far western edge of Europe, none of the traditional trade routes from India & China (overland, either northern or southern; or over sea to Egypt and hence to the Eastern Mediterranean), all of which were in use to varying degrees, would have allowed them to bypass these middlemen.

But for Europeans better situated to take advantage of it, the northern route could be more attractive. The most obvious contender was Russia, and throughout the early modern period the Russian state sought to encourage commerce with Central Asia, leading to a revival of north-south trade routes linking India to Russia across Central Asia via the old "steppe route." Russia therefore did become a major re-exporter of eastern goods to Europe, although we don't know the exact scale of this activity. In the seventeenth century it was certainly much less than the volume of direct overseas trade carried out by European mercantile companies, but it was still far from insignificant. It has been estimated that 20% of all Persian silk exported to Europe arrived there via Russia during this period. While historians of Russia's trade with Europe have typically focused on its native products (and above all furs), it also served as an intermediary for the export of eastern cottons and silks, as well as other goods such as rhubarb, a plant widely used in European medicine. Russia's eastern trade was the topic of a recent study by Erika Monahan which I would recommend if you are interested in pursuing the history of early modern Eurasian commerce (The Merchants of Siberia: Trade in Early Modern Eurasia. Ithaca: Cornell University Press, 2016).