In some cases, you might say medieval European Christians were eager to get into a land war in Asia. Desperate to get into a land war in Asia. King Louis IX of France literally died of dysentery trying to get into a land war in Asia.
I am speaking, of course, of the Crusades.
And even with a success rate of 1/9 (which, to be fair, is higher than the U.S. lethally miring itself in Southeast Asia), crusading never gained the reputation of a classic blunder. In the mid-15th century, enormously powerful Duke Philip of Burgundy threw a massive, multi-day feast to celebrate his taking of the crusader's vow. Did he actually go on a crusade? No. But the ideal was there, and it was shared--by the other men who took the vow at the same feast, and by everyone celebrating them.
I guess that's sort of cheating, though, even if the medieval geographic division of the world into three continents of Asia, Africa, and Europe did put the Near East in Asia ("Outremer"--outre-mer, beyond the [Mediterranean] Sea). So beyond the Crusades, you can make a list of reasons that Europeans didn't want to get into land wars in Asia. Reasons that had nothing to do with not invading Russia in winter.
The first is semi-related to the Crusades: Prester John! Prester John was a mythical Christian king of some land beyond territory ruled by Muslims. For most of the Middle Ages, that meant somewhere vaguely in "Asia" or "India" (India being shorthand for mysterious parts of Asia). The idea was that he would turn out an army to ally with western Europe and help defeat aforementioned Muslims.
So at least theoretically, westerners would not have been interested in fighting other Christians. The evidence itself is a little mixed. France and England certainly never went after Armenia (a Christian kingdom in western/central Asia), but Venetian crusaders were a little too happy to divert from Jerusalem to Constantinople, and plunder and conquer the capital city of a Christian kingdom partially in Europe and partially in Asia.
A third angle to consider is the Mongols. Here I am also going to talk about Eastern European Christians, for the academically critical reason that it makes things more interesting.
With respect to the Mongols, both as fierce nomadic armies and as settled, wealthy khanates, there were multiple situations and motives for Europeans to manage. On one hand, western Europeans' mania for spices and other trade with the far Orient lent itself to diplomatic and trade missions rather than military dreams. The Church's parallel desire to evangelize the Mongols (and perhaps find Prester John) gave another tally mark to a land war in Asia being a classic blunder. A marine-mounted assault, too, since much trade happened via the Indian Ocean.
On the other hand, the people of Hungary in the 13th century might have been very excited for their kingdom to get into a land was in Asia--because it would have meant the Mongols were not getting into a very destructive, murderous land war in Europe.
Best of all, though, comes from the Byzantine Empire in the second half of the 13th century. That was when Emperor Michael VIII dealt with Byzantium's precarious position between Italian city-states and their empires, the Bulgarian Empire, the Turks, and multiple Mongol khanates through marriage. (Not his own.)
Most famous, probably, is his daughter Maria, who married Abaqa Khan of the Ilkhanate to the east-ish. Abaqa, still a follower of the Mongol religion, had a large number of Christians (Christians who had been kicked out of the Catholic/Orthodox Church for heresy centuries earlier, but never you mind that part because the Catholics/Orthodox sure didn't) in his empire. Maria was quite a symbolic and even active leader of them (at least in terms of example and charity). She was also awesome in her own right. Following Abaqa's death and some intrigue at the Ilkhanate court that was not favorable to her, she found ways to justify a return to Constantinople. And her reputation as a Christian working among pagans earned her the name St. Mary of the Mongols.
Meanwhile, Michael chose not to go to war with the Golden Horde Mongols, who were busy invading eastern Europe under their unofficial leader Nogaj Khan. Michael pacified him by arranging Princess Euphrosyne's marriage to Nogaj. Along with a whole bunch of other tribute, but who's counting.
Which does seem to settle the question about land wars in Asia being a blunder.
At least, for now.