Why are there Tatars all over Eastern Europe?

I know the Tatars are natively from the Mongolian Steppe. They were assimilated by Khamag Mongol and became an integral part of the empire, specifically the army. However, there were other tribes in the Empire that Ghengis assimilated. The Naimans, Merkits, Keraites and even the Uighers, who still survive to this day. While the Uighers are for the most part still in their ancestral homeland, the Tatars are now spread all over Eastern Europe and West/Central Asia. From what I can tell they are mainly around the Aral mountains and down along the coasts of the Caspian and Black seas, mainly around the Volga river and in Crimea. There’s even an oblast in Russia named for them (Tatarstan). Why are there so many Tatars still in this area, when the Uighers are still in Mongolia/China, and the rest of the tribes are all practically gone?

Edit: I forgot to mention this. I am keenly aware that the Golden Horde was settled mostly by the Turkic tribes of the Steppe, and that included the Tatars. However, none of the other tribes are around now. That is my question, why is it that only the Tatars seem to have been completely displaced