It's a little.. misleading to call it desertion. The original wording from Carpini's description of the Mongol army was:

When the line goes into battle, if one or two or three or more flee from the squad of ten, all ten are killed; and if all ten flee, unless the rest of the hundred flee, all of them are killed. Briefly, unless they give way together, all who flee are killed. Also, if one or two or more proceed daringly into the fight and the remainder of the ten do not follow, they are killed; and if one or more of the ten is captured and the other comrades do not free them, again they are killed.

The original source was not so much talking about desertion as it was about the units working and moving together. The false retreat, a tactic where the army feigns retreat, hoping the enemy pursues in a more broken/vulnerable formation so that they can counter attack was a steppe favorite, but it only works if well coordinated. So they had punishments to ensure retreats were coordinated affairs.

If I am reading your question right, you seem to be asking if "was desertion so bad that they had to implement this execution order?" When rather it seems that the order was a preventive measure.

See, between 1200-1206 Chinggis Khan completely reorganized the Mongol tribes into the famed decimal system to break up tribal loyalties and the power of the smaller Khans. Originally this was just with the military, and then to all of the tribes. The base unit of this system was the arban, 10 men, who would fight, scout and live together. After 1206, when this was spread to all the tribes, the families of these 10 would then be responsible for providing those ten with arrows, supplies etc. Because of this the arban would have been a very personal thing, very close ties and all that, as it was essentially designed to replace those pre-imperial tribal ties. Betraying this unit then was not just a betrayal of a unit, but people you'd come to see as friends, family, and to know that they would be punished for your desertion would be a strong incentive. Strict rules such as these helped to tie together this new set up and prevent the new nation from unraveling.

As it is, there is a shockingly small amount of desertions and betrayals among Mongol armies during this period. Medieval armies tended to be just lousy with people betraying their commanders. The Jin Dynasty suffered this dramatically in its final years, with deserters founding independent kingdoms (which were in turn swallowed up by the Mongols).

Yet, the Mongols were capable of sending generals riding hundreds of hundreds of kilometres away through foreign lands with total trust that they would return: see Jebe and Subutai's famous campaigns through the Caucasus, southern Russia and back north over the Caspian Sea before remeeting Chinggis Khan (Jebe didn't return because he died, but he if he hadn't he would have). While Chinggis was campaigning in the Khwarezmian Empire his general Mukhali continued to fight in China, gaining titles, wealth and renown and for any other medieval army that certainly would have led to him declaring his independence with his lord on the far side of Asia.

Later on in the century the princes went to war against each other, and there were rebellions, but desertion seems almost entirely unheard of. The Mongols' contemporaries remark with awe the loyalty of the Mongolian warriors, and their willingness to endure every sort of hardship (effectively every single account of the Mongols from the thirteenth century mentions this at some point.)

Does that mean no Mongol ever deserted, and that this executions were never implemented? They probably were at some point. There is absolutely no indication that this hampered a campaign, but know this: while its popular to imagine the Mongols as mindless savage barbarians, they had an incredibly sophisticated military machine, far ahead of most of their contemporaries, an astute understanding of their weaknesses, strengths and their enemies' weaknesses and strengths, and a propaganda machine/ability to spread disinformation like few others. The execution order may have been spread as a part of this to convince others of their fortitude and harshness ( "if we're willing to do this to our soldiers, what will we do to you?" kind of thing) or, if it did occur, I doubt the Mongols or the chroniclers from the Khanates (thinking Juvaini or Rashid al-Din, some of our most reliable and oft used sources) are just not going to mention these things.

Now that is just for desertion. Others may have more on the implementation of this order for other reasons, but I have never read anything which suggests it was every widely used, if at all. Of course, if someone has something which contradicts some of my assertions here I would love to hear it.

Sources:

Paul Ratchnevsky, "Genghis Khan: His Life and Legacy," Peter Jackson, "The Mongols and the Islamic World: From Conquest to Conversion," J.J Saunders, "The History of the Mongol Conquests."