I'll admit I'm not all that familiar with popular history discussions surrounding Timur, but I don't think that I've ever heard the claim that he was a peasant. You are correct however, that Timur's rise to power is often framed as a "Rags to Riches" story, so how true is that claim? Well... I'd argue there is at least some degree of historical basis to that portrayal even if, as you note, it is not completely accurate.

Yes, its true that Timur was a member of the political and military elite of the Ulus Chaghatai; he was from the Barlas clan which was an important tribe in the polity as it was one of major Mongol clans that had been part of the Mongol Empire since its earliest days. So association with the Barlas gave Timur a great degree of political and social clout. Yet this clout can only be understood in the context of the political system in which Timur lived. During the time of Timur the political situation in the Ulus Chaghatai was incredibly fluid; the authority of the Khan was basically absent, and whoever held the most military power was able to control or prop up a Khan as puppet. However the military power necessary to control the Ulus Chaghatai was also difficult to maintain, as the loyalty of tribes and military groups would often shift very easily. This meant that wars for power and military conflicts within the Ulus Chaghatai was almost constant. Yet, while battles were very frequent they were also fairly bloodless; as soon as battles began to shift in one side's favor the troops of the other side would disperse or defect to the other side. This meant that political control and power was always shifting hands during this period. Anyways, what would happen to the losers in these battles and skirmishes? Well, they took up a practice that is termed by scholars as "Qazaqliq"(Also the origin of the term Kazakh and Cossack), where they would basically live as brigands in the steppes and countryside. This was a life of incredible poverty and harshness, but it also enabled members of the military elite to regroup and rebuild their political and military power. If one was a good enough 'Qazaq' and built up a large degree of notoriety, they would be able to build social and military clout and attract very loyal followers to support them, essentially building an army from the ground up, and it enabled these losers of the initial conflict to once again rise up and have another chance to take power.

Timur spent much of his early life as a Qazaq, living life essentially as a brigand and it was during this period that he received his famous injuries. Yet, even though this was an incredibly harsh and difficult way of life he was not a nobody in this period, the purpose of being a Qazaq was to build an armed following and retake power and this endeavor was certainly helped by Timur's background and familial connections. Yet even with this, Timur's success was not guaranteed and his seizure of power was certainly not an easy task for him.

TLDR and to more directly answer your Question: Popular history discussions of Timur tend to lack an understanding of the social context of the Ulus Chaghatai during Timur's rise to power, which leads to a simplified understanding of what Timur's early life was like and leads leads to assumption that Timur started worse off than he was in reality. He did have a rough time taking power, but it wasn't that rough as some may claim.

The story tha got an arrow while stealing cattle also semm ficktiv as the russian scientist who exermined his corpes proofed that his condition was the result of Bone-Tubercolosis.

I have never heard of this claim until now actually, and looking it up I can't seem to find any academic sources that verify this (That he got his lameness from tuberculosis as opposed to an arrow wound), so I'd actually appreciate a source for this.

Sources:

Manz, Beatrice Forbes. The Rise and Rule of Tamerlane . Cambridge: Cambridge University Press, 1989.

Lee, Joo-Yup. Qazaqlïq, or Ambitious Brigandage, and the Formation of the Qazaqs, (Leiden, The Netherlands: Brill, 2015)

Gommans, Jos. " The Warband in the Making of Eurasian Empires". In Prince, Pen, and Sword, (Leiden, The Netherlands: Brill, 2018)