No, nothing is missing or overlooked in "trew Law" regarding apostolic succession. The text simply does not provide what this unreferenced part of the Wikipedia article claims.
What James writes is that the kings are sent to their people by god, and the link to apostles is only found in Paul being used as witness 'that people should obey their masters', as god would have sent them for reasons.
Kings are called Gods by the propheticall King Dauid, because they sit vpon GOD his Throne in the earth, and haue the count of their administration to giue vnto him. Their office is, To minister Iustice and Iudgement to the people, as the same Dauid saith: To aduance the good, and punish the euill, as he likewise saith: To establish good Lawes to his people, and procure obedience to the same, as diuers good Kings of Iudah did: To procure the peace of the people, as the same Dauid saith: To decide all controuersies that can arise among them as Salomon did: To be the Minister of God for the weale of them that doe well, and as the minister of God, to take vengeance vpon them that doe euill, as S. Paul saith. And finally, As a good Pastour, to goe out and in before his people as is said in the first of Samuel: That through the Princes prosperitie, the peoples peace may be procured, as Ieremie saith.
[…]
And what was Nero to the Christian Church in his time ? And yet Ieremy and Paul (as yee haue else heard) commanded them not onely to obey them, but heartily to pray for their welfare.
– James: The Trew Law Of Free Monarchies: Or The Reciprock And Mvtvall Dvetie Betwixt A Free King And His Naturall Subjects. edited by Charles Howard McIlwain
The WP page seriously conflates this with the doctrine of apostolic succession being under discussion at the time in Scotland, for the church and regarding reformation, concerning the bishops, to which James had a very pronounced opinion:
In Basilicon Doron James characterised the Scottish Reformation as ‘inordinate’ and ‘not proceeding from the prince’s order’. Many of his actions, such as his steady re-establishment of Scottish episcopacy after 1596, were designed to lead the kirk into greater harmony with the English church as well as increasing royal control.
James believed that monarchy was divinely ordained, and he held that the apostles themselves instituted bishops in the early church.
He supported the English Church as he found it under Whitgift: largely Calvinist in doctrine but episcopal in organisation. The Scots, however, regarded their kirk as setting a purer example of Reformation as well as contributing to their distinctive nationhood. In 1604 in deference to Scottish anxieties, messengers from James assured the commissioners of the General Assembly that a union of churches was not part of the agenda of Union. The Parliament held later that year in Edinburgh passed an act excluding the kirk from the scope of the Union commissioners. Even so, there were rumours around the time of Bancroft’s promotion to the see of Canterbury in October 1604 that the king would have liked to make him Primate of Great Britain, not just England, but held back for fear of offending the Scots.
– Pauline Croft: "King James", Palgrave Macmillan: Basingstoke, New York, 2003.
But that means that the church itself is meant in his opinions on apostolic succession, not the kingship – although they are related in struggling for power and 'purity'.
(– Cf Kenneth Fincham & Peter Lake: "The Ecclesiastical Policy of King James I", Journal of British Studies, Vol. 24, No. 2, Politics and Religion in the Early Seventeenth Century: New Voices, 1985), pp. 169-207.)
The introduction to a collection of his political writings makes this more clear:
Of these opinions the doctrine of legitimism was among the most important. Allegiance, he says, is due not only to the reigning king, but also to his
"lawfull heires and posterity, the lineall succession of crowns being begun among the people of God, and happily continued in diuers christian commonwealths … For, as hee is their heritable ouer-lord, and so by birth, not by any right in the coronation, commeth to his crowne, it is a like vnlawful (the crowne euer standing full) to displace him that succeedeth thereto, as to eiect the former: For at the very moment of the expiring of the king reigning, the nearest and lawful heire entreth in his place: And so to refuse him, or intrude another, is not to holde out vncomming in, but to expell and put out their righteous King."
"But if God giue you not succession," he warns his son, " defraud neuer the nearest by right, whatsoeuer conceit yee haue of the person: For Kingdomes are euer at Gods disposition, and in that case we are but liue-rentars, lying no more in the Kings, nor peoples hands to dispossesse the righteous heire."
In the speech from the throne opening his first English Parliament, James insisted upon the same point, and could have found little to quarrel with in Parliament's answer, "That immediately upon the Dissolution and Decease of Elizabeth late Queen of England, the Imperial Crown of the Realm of England, and of all the Kingdoms, Dominions and Rights belonging to the same, did by inherent Birthright, and lawful and undoubted Succession, descend and come to your most excellent Majesty, as being lineally, justly and lawfully, next and sole Heir of the Blood Royal of this Realm."
James, of course, was not unaware that his right though divine and heritable, could be traced to a historical beginning. How was he then to distinguish between his ancestors' acquisition of the Crown from which all his own rights flowed, and the mere de facto sovereignty of any usurper?
Here the analogy of private property again proved useful. Conquest is to be distinguished from usurpation; the conqueror is in much the same position as one who acquires title by occupatio; and quod … nullius est, id ratione naturali occupanti conceditur. But Scotland, and England as well, was conquered by James's ancestors. Thus their right to the realm is nothing less than an absolute ownership, and neither the people nor anyone else can have any rights in what is solely theirs; neither can the people by laws of their own making interfere with the owners' enjoyment of what is theirs alone."
— same book by Charles Howard McIlwain
