The definition of "as of right" in Your Dictionary is
(law) by means of a legal entitlement, rather than through extenuating circumstances.
The same source quotes Webster's New World Law Dictionary as defining the term this way:
Description of a court action that a party may take without permission of the court, as opposed to requiring leave of court.
Perhaps the easiest way to interpret the phrase "as of right," therefore, is as having the meaning "as a matter of right."
Various early occurrences of the phrase seem to reinforce this reading. For example, Journals of the House of Lords (1610) has this instance:
That whereas the House of Commons have already, among their Grievances, preferred a Petition to His Majesty, as of Right and Justice, That the Four English Counties may have a Trial by Law, concerning their Inheritance to the Common Laws of this Realm, and so to be exempted from the Jurisdiction of the President and Council of Wales (a Matter wherein the whole Realm is deeply interested) ; notwithstanding, upon Occasion of this great Contract, the House of Commons doth humbly petition His Majesty, as of Grace, that, without further Suit, Trial, or Trouble, those Counties may be restored to that their ancient Right, the same being no way prejudicial to His Majesty's Honour in Point of Sovereignty (as we conceive) ...
And from Encyclopædia Americana (1830):
In all criminal cases, the accused is entitled, as of right, to the assistance of counsel in his defence ; and this right, also, is generally secured by the state and national constitutions of government.
The sense of "as of right" in these examples seems consonant with the sense of "as a matter of right" in (for example) The Tryal of Thomas, Earl of Macclesfield in the House of Peers, for High High Crimes and Misdemeanors (1725):
My Lords, we do not ask this as a Matter of Right, but as a Matter of Indulgence. I own the Rule that the Gentlemen of the House of Commons have laid down is a general Rule for the Government of Evidence.
And the same seems true of this excerpt from Edmund Burke, The Revolution in France, second edition (1790):
It will always happen, when a thing is originally wrong, that amendments do not make it right, and it often happens that they do as much mischief one way as good the other : and such is the case here ; for if the one rashly declares war as a matter of right, and the other peremptorily with-holds the supplies as a matter of right, the remedy becomes as bad or worse than the disease. The one forces the nation to a combat, and the other ties its hands : But the more probable issue is, that the contrast will end in a collusion between the parties, and be made a screen to both.