This is an interesting question because it forces us to examine not only the creation of historical myths but also the way that those myths were actively used by kings and their descendants as propaganda and justification. Such processes were certainly at play in medieval Scotland, and indeed, historians are still working to unravel the truth about the origins of the kingdom of Dál Riata and its rulers today. This is because the only real sources we have that discuss these figures are medieval Irish and Scottish chronicles, which were themselves often written as instruments of propaganda, officially commissioned or sponsored by particular rulers, bishops, or nobles, or dedicated to them in order to curry favor. Thus, leaving the fanciful myths aside (for instance, that Britain was conquered by the king of Albania and Brute of Troy, a descendant of the Trojan hero Aeneas), the "facts" contained within these sources must be critically assessed for how factual they actually were and what motivated the person writing these accounts to do so.

But, according to surviving medieval sources like the Duan Albanach (Song of the Scots), Dál Riata was founded by the three sons of Erc of Dalriada, who himself claimed descent from the famous Érainn king Conaire Mór. Erc's sons, Fergus Mór, Loarn, and Óengus, expanded the borders of the Gaelic kingdom of Dál Riata to include the western seaboard of Scotland around 500 CE. This descent, as it is presented in medieval sources, is important. According to the Old Irish saga, Togail Bruidne Dá Derga, Conaire Mór was himself semi-divine, having been conceived by his mother after a strange man flew into her room in the form of a bird. (Such devices were frequently used in classical antiquity as well--consider the number of Greek heroes who were fathered by Zeus in various animal forms.)

While the kingdom of Dál Riata flourished in the western Highlands and Islands of Scotland, a rival kingdom of Picts occupied the eastern half of the country. Similar to the political structure of Ireland, Pictland was subdivided into as many as seven smaller kingdoms with one "high king" or "overking" ruling over the others. Although the Pictish Chronicle records the names of fifty-three high kings of the Picts before the succession of Cináed mac Alpin (Kenneth MacAlpin), the first "king of the Scots", and the Annals of Ulster and the Annals of Innisfallen mention by name several "kings of Fortriu" (the name of one of the Pictish subkingdoms that frequently dominated the others and therefore, furnished many of the Pictish high kings) none of these sources say much about how the first king of Pictland first rose to rule over his kingdom. What we do know, however, is that succession within Pictish society was apparently matrilineal. When Cináed mac Alpin, who was already king of Dál Riata, became king of the Picts in 843 (and thus the first king of a united Scottish kingdom), he based his claim on the fact that his mother was a Pictish princess, as was his paternal grandmother.

Yet, Cináed mac Alpin was given the posthumous nickname "An Ferbasach" ("The Conqueror"), having been characterized by the Chronicle of the Kings of Alba (a later medieval chronicle dating probably to the thirteenth century) as the conqueror of Pictland, a claim later echoed by the fifteenth-century Scottish chronicler Andrew of Wyntoun. While there is no contemporaneous evidence to suggest that Cináed's succession was especially violent, or involved the apparent "destruction" of the Picts as suggested by the Chronicle of the Kings of Alba, this was the myth that sprung up around the figure of Cináed mac Alpin. More recent scholarship has argued against the historicity of this narrative though, as it wasn't until Causantin (Constantine II), Cináed's grandson, that these kings began to refer to themselves as "kings of Alba" rather than as "kings of the Picts".

The House of Alpin ruled the kingdom of the Scots for 117 years before Malcolm II died without male heirs. Before his death, he designated his grandson, Duncan, son of his daughter Bethóc and her husband Crinán, hereditary lay abbot of Dunkeld, as his heir. Duncan was killed in 1040 by Macbeth, who became king without much resistance. Little is known about Macbeth's mother; however, she may have been another daughter of Malcolm II, which would have made Macbeth and Duncan cousins, and which may account for the lack of resistance to his proclaiming himself king. Macbeth was succeeded by his designated heir, his stepson Lulach, but Lulach was perceived as weak and ineffective and ruled only for a few short months before he was killed by Duncan's son, Malcolm III, who became king in 1058. Malcolm III was succeeded by his younger brother, and then by each of his sons in turn, two of his great-grandsons, and then his great-great-grandson and great-great-great-grandson before this final king, Alexander III, fell off his horse while riding to visit his new queen in the middle of the night and broke his neck, leaving no heirs aside from a three-year-old granddaughter in Norway whose ship sank on her way to Scotland to be crowned.

Thus came the succession crisis that sparked the Scottish Wars of Independence and eventually ended with Robert Bruce claiming the crown by virtue of the fact that his great-great-great-great-grandfather on both sides had been David I, Malcolm III's youngest son. Robert I was succeeded by his son, David II, who died without heirs and named his sister's son, Robert Stewart, his heir. The House of Stewart/Stuart ruled Scotland (and England and Ireland) until 1714, when Queen Anne died childless and was succeeded by George I, whose grandmother had been James VI/I's only daughter, Elizabeth.

Usually what you see (to the extent we can say "usually", which is... not all that far, given "medieval" encompasses a lot of time and place with a wide amount of variety) is that a royal house, dynasty, or line is created when someone (or really, some family) already very highly placed with some vague-to-good claim on the throne manages to push out a weak existing king or win through any of several means a vacant(ish) throne. On occasion the "claim to the throne" is simply "conquest", but even that requires you to be already a king or nearly so.

Let's take England. The House of Wessex ruled Wessex and England (such as it was) until such time as the Danish king Sweyn invaded and simply took over. The Danish line and the Wessex line went back and forth a few times until Edward the Confessor, a Wessex king, died with no heir. Several people took a shot at the throne: a high-ranking Anglo-Saxon earl with blood ties to both the Danish and Wessex royals, a king of Norway with (supposed) ties to the previous Danish rulers, a (nearly-sovereign-already) duke of Normandy who was a cousin to the previous king, and a fellow who was technically next in line by blood but who had only recently been discovered in Hungary.

So this is a pretty good lineup of potential replacement kings: local high nobility, a foreign king, a neighboring noble, and a blood relation. In this case, they all fight it out and William of Normandy wins. He's vaguely related (first cousin once removed), maybe the king made a promise to him, but moreover he has a nice existing power base and a victorious army marching through the country.

And this is typically how it goes, except the means and outcomes vary. William's direct line didn't last long, and when his grandson died without male children, they tried to place a daughter (Matilda) on the throne, but a nephew invaded and started a civil war until they compromised by allowing the daughter's issue to be heir. Matilda's husband happened to also own large portions of France, and thus united them with the throne of England. The Angevins/Plantagenets, thus begun through marriage, would be quite long-lived.

Such was the power of descent, and the descent of power, that nobody became king of England that wasn't a stone's throw away from it already.

Your wonderfully meritocratic version is really quite rare; I'm sure there are such elsewhere (and I'd love to hear about them!), but the only major example I can really think of is Oliver Cromwell. And he's not medieval, nor technically royal. But he was, at least, not already practically royalty.