The best part of this is: trust me I guarantee Tolkien knew this much about the Common Law (English mediaevalists end up knowing ridiculous amounts about both Common Law and mediaeval Catholicism whether we want to or not), and indeed if you look at the text, this was relevant to the story.
It’s part of the reason that Sauron is as terrified of Aragorn’s potential claim on the Ring as he is of Gandalf’s or Saruman’s or Galadriel’s - if not more. Because in Middle Earth this shit matters. This is a world where a broken oath will literally bind your unhappy restless soul to the earth in spite of the dictates of the literal creator of the universe (who designated humans as Passing Beyond The World when they die). This is a world where a damn oath is responsible for Everything That’s Wrong With The First And Second Ages.
Oaths, ownership, duties, rights, things owed and owing: this shit matters.
And sure Aragorn is also direct line from Lúthien, but so is Elrond, and so are Elrohir and Elladan. So is Arwen. But what none of them have that Aragorn has? Is a rightful claim to ownership of the Ring.
So much of what Aragorn spends his time in the second and third volumes doing is Establishing Claim - establishing that everything that Isildur owned, he now owns. Why? Because it means he has power that is absolutely needed. “Isildur’s Heir” isn’t a woo-woo floofy-high-concept thing: it’s a literal matter of rights, duties and authority.
When he takes the Palantír from Gandalf and uses it, his companions are aghast, but he reminds them that he has both the right and the strength to use it - and the Right is actually important. Saruman was, face to face, stronger than Aragorn (never doubt that) and Sauron completely pwned him, but Saruman had no right to the Seeing Stone, no more right than Pippin.
But the Palantíri belonged to Aragorn: he’s not only Melian’s ever-so-great-grandchild, he’s also Fingolfin’s ever-so-great-grandchild, and since the Fëonori died out with the poor Ringmaker, the only competition Aragorn could have for ownership of the Stones are Galadriel and Elrond. (And that’s only if you are going right back to the maker-rights, and ignoring the establishment of the Stones as the property of Elros’ line rather later).
It matters. It changes how power works and doesn’t work. Aragorn’s status as the Heir is in fact grounded in these ideas, which play a hugely powerful part (in fact the fight over who rightfully owns the Silmaril Beren and Lúthien brought out of the dark is part of the bloodshed that makes it so that in the end the Silmarils themselves actively reject the last two living sons of Fëanor, negating their claim). Because Aragorn is the rightful inheritor of everything Isildur ever had, he can use the Palantír. Because he is the rightful inheritor of everything Isildur ever had, he can summon the Dead. And because he is the rightful inheritor of everything Isildur ever had, he stands equal to two of the Ainur, to the oldest member of the Trees-blessed Noldorin royal house, and to his own much more powerful (straight up) relatives as a potential claimant of the Ring.
And that is why Sauron is willing to take the chance to catch Aragorn, and (he thinks) ensure his capture, rather than attacking him earlier on when there’s a chance that (even if Aragorn can’t possibly WIN) he could still escape and then bide his time before the next Ring-War and learn to use the damn thing.
But. It’s also important when it comes to Frodo.
Frodo uses the Ring twice, and lays open claim once. Both of the times he uses it are on Sméagol, both times overwheming him and in the second case cursing him (“if you ever touch me again you will be thrown into the fire”). We get both moments from Sam’s POV, where the physical reality of Frodo is replaced by an image of him as a much larger figure, alight from the inside, robed in light, and with a “wheel of fire” at his breastbone.
Frodo does not have any genetics (so to speak) more special than any other hobbit. It’s not like Aragorn vs most humans, where there’s actually a legit difference because most humans were not, at that point, descended from a Maia. Frodo’s just this guy.
The only thing that’s really special about Frodo in terms of the Ring is that, like Aragorn, he’s the other person who has a viable claim. It would, as it were, have to go to the judges to figure out whose claim is better.
And this is why in the moment that he claims the Ring, in the Mountain, Sauron is fucking terrified. It’s why he drops everything else, even the issue of trying to keep his mindless drone-fighters going, even the maintenance of his actual control of weather, of light, of whatever fight he and Gandalf have going, to get his best servants back to the Mountain now now now now.
Because Frodo having an actual rightful claim on the Ring means he can, in fact, use it. Not well, which is why Sauron can paralyse him for that moment it takes for Sméagol to strike (and carry out both Frodo’s demanded oath - “save the Precious from Him” - and his Curse - “if you touch me you will be thrown in the fire” - at once), but he could. This tiny little person is a threat to Sauron, in the heart of his own home, because he has the right to have and use this Ring.
The tricky thing about Tolkien is that whatever his flaws (and he has many), the one thing he’s never unclear of is that the concept of right and might are actually separate. Just because you are strong enough to do or take a thing doesn’t mean you have any right to do it; and just because you aren’t strong enough to enforce your right, doesn’t mean it goes away.
