By XVIth-XVIIth century, the demographics and the technological disparity simply did not favor another steppe empire. Some time by the end of the XVth century, the demographical balance has shifted irretrievably towards the settled, sedentary, civilized areas, and technological development taking place in such areas meant that steppe armies were hopelessly outclassed. There is a reason Tamerlane's conquests were the last large-scale conquest by a "steppe army", and even then, it was stretching it. Babur's later conquests were on much smaller scale, and using an army that was less like the Mongol armies than those of his predecessors.
To put it simple, a XVIth-XVIIth century steppe army, whose strength is based on extreme mobility, may face problems when trying to stand against contemporary armies that rely less on wild cavalry charges, and more on superior firepower, such as cannon, firearms, etc. To put it simple, Ghenghiz Khan's army has nothing to answer even the XVIth century cannon with, and if it tries to compete with cannon-equipped armies on their own terms by creating their own artillery, they would lose their greatest advantage, which is mobility and coordination. By creating an obvious sedentary position, you have to create a battle plan that minimizes your mobility advantage, which makes you just another XVIth century army. And by XVIth century, there were simply not enough Mongols to form a large, well-trained military force capable of standing toe-to-toe with Chinese, Russian, Ottoman, or Western European armies.
Finally, let's consider who such an would-be Second Mongol Empire would face. China is probably the easiest picking, due to much internal unrest and fractious nature of the country at the time. But even then, Mongol army would be outnumbered, and the Chinese might prove more of a dangerous opponent this time around, if only due to being more technologically advanced, with higher population, and with the experience of having fought the Mongols before (even if it was three centuries, I would imagine the first thing any would-be Chinese general would do is do his research, if he has half a functioning brain).
Then, there is the Timurid/Moghul Empire in India, who can claim at least some descent from Ghenghiz Khan, who are well entrenched, familiar with steppe tactics to an extent, and, I believe, sufficiently well equipped with reasonably modern (for the time) weaponry, and have a large, well-trained army. Even if the Second Mongol Empire bypasses them completely, and goes further West, things don't get easier. The Ottomans are near the peak of their power, their army is well-supplied, well-equipped, and armed with some of the most modern weapons of the time. Not to mention that it was before the worst of the Ottoman military leadership crisis set in. So the Ottomans will not be a pushover for the control of the Middle East.
Then, Russia. Under Ivan IV and his predecessors, Russia has modernized its society and warfare quite a bit, and will be more than a match for the steppe army. Especially with gunpowder weaponry, experienced officer corps, and no shortage of potential recruits. A pure steppe army has no chance against that combination, and the post-Mongol-state armies Russia did fight in OTL ended up being organized much closer to Russia's own army - such armies have a disadvantage in that they are basically just another "normal" army, requiring high enough population to draw recruits from, reasonable manufacturing capacity, good supply lines, and access to the materials that are needed to create a XVIth-XVIIth century war machine. Most of which is not available in Mongolia, and while it IS available in China, it also makes it much harder to conquer by the Mongols in first place.
Final verdict? Tamerlane was the last gasp of the "steppe armies", and even then much of it was because he was faced with less technologically advanced areas that have already been devastated by previous invasion routes, or the areas whose military technology and doctrines were not yet completely developed. By 1500, Tamerlane's army would have had an incredibly hard time accomplishing half of what it did... by 1600, it would not have managed to do much damage at all against the contemporary forces of the time.