Karna was elder brother of Pandavas. As per the ancient Indian monarchial system, eldest son has a right over throne of his father. Vrishketu was the only son of Karna survived after Mahabharat war and real owner of throne. So why he didn't get the throne?
First by defintion, Karna was a Kanina and could be made king
Manusmriti 9.172 says
pitṛveśmani kanyā tu yaṃ putraṃ janayed rahaḥ | taṃ kānīnaṃ vadennāmnā voḍhuḥ kanyāsamudbhavam || 172 ||
If a maiden secretly bears a son in her father’s house, that son, born of a maiden, should be declared as ‘maiden-born’ by name, and to belong to the man who marries her.—(172)
Lord Krishna, in Udyoga Parva, says the following to Karna
It is said by those conversant with the scriptures that the two kinds of sons called Kanina and Sahoda that are born of a maiden, have him for their father who weddeth the maid. Thou, O Karna, hast been born in this way. Thou art, therefore, morally the son of Pandu. Come, be a king, according to the injunction of the scriptures
So there is no doubt that Karna and his descendants were eligible.
Now why wasn't Vrishaketu made the King ? That is because there was no such person in Mahabharata.
The only reference of Vrishaketu was found here in KMG Mahabharata
That large force of the Easterners looking like the fleecy clouds of autumn 3 was (besides) protected by the chief of the Angas (Karna's son Vrishaketu) and Kripa endued with great energy.
But this seems to translation error. The sanskrit verse (see number 28) only says "chief of Angas" - who was Vrishasena (son of Karna), who died in the war
tad aṅgapatinā guptaṃ kṛpeṇa ca mahātmanā śāradābhracaya prakhyaṃ prācyānām abhavad balam