Why is it that a powerful Russian state based out of Kiev never re-emerged after the collapse of the Mongol empire despite having been the most prominent Rus' city prior to that?
Also why it is that Kiev was sacked during the Mongol conquest and yet other large cities such as Moscow and Novgorod managed to escape that fate?
Firstly, Moscow was never a large city prior or during Mongol conquest. And it was sacked. Secondly, Novgorod avoided the fate because it was far enough away to the Mongols to not bother to go there. Novgorod accepted the Mongol rule anyway. Thirdly, there was no 'powerful Russian state' based out of Kiev prior to the Mongol conquest. While Kiev principality was the richest and most populous of all Russian principalities, it was still relatively small. United Rus ceased to exist more than half a century before Mongols came.
The reason why Kiev never recovered was not even tied to the Mongols. The source of the Kiev wealth and power was trade along Dnieper river from the Baltics through Novgorod and to the Constantinopole and then Levant and further east. But this trade 'dried up' with the decline of Byzantine Empire and because Crusades reestablished alternative trade route with the East through Mediterranean sea.
Because of that Baltics-Volga-Caspian Sea became a main trade route in the Rus lands. Novgorod controlled the Baltic part of the route still but Kiev was now out of the way. Because of that center of power gradually switched to the Vladimir and then to Moscow. It would happen even without Mongol invasion just more slowly.
The sources for the post are various lectures by historian Klim Zhukov (unpublished) and Khrustalev's work "Rus and Mongol invasion" (Хрусталев Д. Г. Русь и монгольское нашествие (20-50 гг. XIII в.). — Спб.: Евразия, 2015)