My gut feeling is that the Mongol conquests (and subsequent consolidation into a brief but very large empire) were a once-in-an-eon historical occurrence, and that kind of thing is very easy to screw up. Subutai was quite possibly history's most successful general, but the steppes have forged great generals for ages - the trick is administration.
Temujin, himself extremely openminded, was capable of forging a state - not a lasting state, mind you, but one which survived its leader, had a unified code of law, and was capable for a time of administering vast groups of nomads and settled peoples. It was not capable of doing so particularly well, which led in no small part to the revolts which caused its downfall, but it did it better than many other nomadic peoples who won military supremacy. Can Subutai do the same successfully?
Subutai replacing Temujin would require Temujin himself to successfully get rid of hereditary succession in favor of some kind of meritocracy. This is itself a huge change, perhaps one with greater effects on the Mongol Empire than Subutai's rule, and may or may not be a good thing - on the one hand, hereditary rule provided little stability OTL, when Genghisid siblings and cousins battled a ton for the throne, starting with Genghis' death and questions of Jochi's legitimacy.
On the other hand, a system of government where any ambitious general can become king itself encourages civil wars (which were common anyway) and prevents a very useful form of legitimacy, in the form of direct male-line descent from a heroic leader figure. Even if the Kurultai was given the authority to choose a non-Genghisid khagan by Temujin himself, its decisions were often rebelled against when other powerful figures saw it in their advantage to do so, and it was often unable to enforce its will, especially in the distant corners of the empire. I think the best you can hope for through Subutai and such a system is stability as bad as OTL's, but with more competent people (starting with Subutai) guaranteed to be at the helm. However, there's a real risk that doing this would accelerate fragmentation - witness the fate of Alexander's empire.