I have no idea how, say, Chinese or Western historians view the history of Mongols after fragmentation of the empire but some readings from the Russian ones have been somewhat confusing for me since, according to the most prevalent opinion in Russian historical circles (including Roman Khrapachevsky, probably the most respected specialist in this area of research here), Mongols pretty much never suffered defeats aside from 1-2 minor engagements and were, more or less, relatively the best fighting force in the history of the mankind. Defeats and collapses suffered by Mongolian states after defragmentation of the empire do not count since those were not "true Mongols": existing outside of their ancstors' way of life (constant hunting and horse riding since childhood, perfected archery skills, harsh life with little to no comfort), they quickly grew soft and thus were more easy for the sedentary people to defeat.
Many very strong claims are usually brought up, including the fact that horse archery is the best possible way of waging warfare before mass produced artillery and gunpowder weaponry. According to Khrapachevsky, essentially, Mongols managed to defeat everyone they came across as long as they remained united and had "original" troops coming from the steppes, and every single time they were defeated (aside from ambush here and there) it happened long after they became weak warriors. Every time Mongolian States managed to somehow retain their archery and tactical skills they came out on top again and again (Tatars defeating European enemies at Vorskla in 1399, Timur destroying sedentary Turks at Ankara in 1402), and only "softer life" saved the rest of the world from utter destruction. Some even go as far as claiming that European/Indian/Japanese/etc would've been entirely hopeless against Mongolian onslaught even in 15-16-17 century, since Mongolian tactical and fighting capabilities were that good. I remember a citation from the book akin to "Mongolian Empire existed for 10 times lesser time period than Roman Empire yet managed to capture 10 times as many lands".
What do other historians (especially from other countries) think of these claims? Were Mongolian/Tatar descendants indeed as awful as it is often brought up?
Many Western historians would, I think, certainly dispute the claims of "real" Mongol invincibility after the decline of the Mongol Empire—at least past the fifteenth century. You bring up Timur's campaigns against the Ottomans, but not his wars with Toqtamysh in 1391 and '95—or, even more relevantly, the failure of the Chaghatay Khanate to maintain a hold on Transoxiana earlier in the century. (It's important to remember that while Timur came of nomad stock, his power really rested on the sedentary population of Transoxiana—and if you can make the argument that his troops were more nomadic than those of the Ottomans, there were states still more nomadic than his.) Quite a bit later, and on the China side, the Battle of Jao Modo in 1696 showed that the Qing Empire was decisively able to hold its Inner Asian frontiers.
For a long time, it was traditional in Western historiography to ascribe the decline in Mongol power to the adoption of Buddhism in Eastern Central Asia—which, so the story goes, made pacifists out of men formerly accustomed to hard and violent lives, and moreover took a large number of men out of the potential fighting force to become lamas. (This idea is somewhat similar to the Russian one you mention, since it's founded on the principle that if the Mongols weren't constrained by their religion, the world would be constantly in danger of another Mongol invasion.) This theory was, however, attacked by Owen Lattimore in his Inner Asian Frontiers of China (1940) and has long since fallen by the wayside. Certainly, by the nineteenth century, Western travelers in Mongolia did not see any trace of the Mongols' world-conquering ancestors in the men they encountered—the Travels of Évariste Régis Huc, probably the most famous piece of Mongolia travel-writing from the period, views the Mongols rather as objects of pity exploited by their canny Chinese neighbors. (This was compounded by the advent of race science, which put the Mongols on a decisively lower footing vis-à-vis the West; on this, and more generally later Western views of the Mongols and Tatars, see the not-exactly-balanced Mongols in Western/American Consciousness by Kevin Stuart.)
That said, of course, many historians today also hold that the Mongol states that adhered closer to nomadic tradition were stronger than their more sedentarized neighbors. (The key point here is that even if they were stronger, they were certainly not invincible.) In an article on "The Decline and Fall of the Mongol Empire," David Morgan argues that it probably wasn't a coincidence that the strongly sedentarized Yuan and Ilkhan states fell long before the more nomadic Golden/Great Horde and the Chaghatay Khanate. He certainly seems to echo some of the Russian points you mention: "[a] competently led and adequately armed force of Central Asian cavalry archers was, other things being equal, unsurpassable on the field of battle (however useless it might be in siege warfare), until the development of efficient battlefield artillery and handguns." But Morgan also concedes that adherence to traditional norms may have also been a weakness, especially in the case of the very nomadic Chaghatay state, which lost control of its rich Transoxianan territories in the mid-fourteenth century.
In short, the consensus is certainly that Mongol military power did degrade after the middle of the thirteenth century, though the question on why and how it did so has long been a matter of discussion among Western historians. The theories I've mentioned so far were held by people concentrating specifically on Central Asia and/or its immediate neighbors; suffice it to say that there is another tradition of world-historiography that seeks to place the decline of steppe power in the context of global shifts in circumstances, but for reasons of space I won't treat it here.