If God wanted Kaliyug to start immediately after Mahabharata and eliminated (divine offsprings ) them in the name of war, left this world for the ordinary , and for kali to take over...then how Pandavas ruled for 35 years followed by Parikshit and his dynasty for next 1000 years?

Was Raja Parikshit an ordinary person ? (I guess no as Arjuna was his Grand Father)

Kshatriyas have natural tendency to uphold Dharma [as per veda] , upon that if kshatriyas are from the lineage of devatas [as was in pre mahabharath era , almost entire devatas were born in kshatriya clan] , these would be spread in every part of the world and naturally their offsprings will also be strong and capable of halting speedy spread of kaliyuga. that way if there are everywhere mighty kshatriyas then Kaliyug will not spread at all .
so except Parikshit[devata kshatriya] everywhere else ordinary were ruling ,
even for Parikshit Kali tricked him to get five places . during 35 years of Pandava rule kali and bali created havoc in the periphery of the earth for[ this was possible beause they had penetrated successfully in those countroies as rulers were weak.

Bheemasena attacks them and brings them in captivity to Yudhisthira , Yudhisthira asks why havoc in his kingdom,
kali says whose kingdom " this is my kingdom according to yuga dharma " you are going against yugdharma.

Yudhisthira happy with his answer grants him pranadana and hey are captivated for 1000 years in a trunk , thus pandavas ruled 1000years . when kshemak out of curiosiy releases them , his kingdom fell down .

thus Srikrishna allowed Kaliyug to grow. and only one race Pandavas could still check its growth.