In the medieval Arthurian legends, Morgaine la Fay is not the woman who tricked Merlin and imprisoned him. There is no cave of ice. In one version (“The Prophecies of Merlin”), Merlin was imprisoned by her in a cave where his spirit could afterwards be found. In another version (“The Post-Vulgate Merlin”, summarized in Malory's “Le Morte d’Arthur”) Merlin was imprisoned in a tomb which was part of an artificial house built under the the ground, where Merlin died. in yet another version ("The Vulgate Merlin”), Merlin was imprisoned in an invisible tower and seemingly does not die. The woman responsible is the Lady of the Lake, not Morgaine la Fay who is an entirely different person.
Morgaine was not the mother of Mordred in any extant medieval text. The mother of Mordred is another half-sister of Arthur sometimes named Morcades or Orcades or Norcades or Sangive or Margause or Belicent. She is sometimes Arthur's full sister named Anna. She is the mother of Gawain by her husband Lot and mother of Mordred also. In earlier stories Lot is also Mordred’s father, but in the later cyclic prose romances Arthur is Mordred's father.
The idea that Morgaine is Mordred’s father is pure invention by some modern novelists. Also, in medieval romances Mordred is never known as the “Black Knight”.
In Robert de Boron’s medieval work “Merlin’', Morgaine is said, in a discussion of the daughters of the Duke of Tintagel and Ygraine, to have been begotten “en baste”, that is begotten in bastardy, begotten as a bastard. It is not said whether this means that she is an illegitimate daughter of the Duke of Tintagel or an illegitimate daughter of Ygraine, but since earlier Ygraine is praised as being very loyal to her husband, it is more likely that Morgaine is the Duke of Tintagel’s bastard daughter. Also, a casual mention of a bastard daughter would, taking into accounts medieval customs, be far more likely in referring to the daughter of a man than to the daughter of a woman. As Arthur is begotten on Ygraine by Uther Pendragon, Morgaine would therefore be his step-sister, not half-sister.
But in the “Prose Lancelot”, Morgaine is said to be the daughter of the Duke of Tintagel by Ygraine, and thus Arthur's half sister.
In some other accounts Morgaine is not Arthur’s sister at all, and is shown to be active in long before the beginning of Arthur’s reign. In the “Perceforest” for example, she appears in Britain long before the coming of Joseph of Arimathea to Britain. In the romance “Huon of Bourdeau”, the fairy king Auberon is son of the famous Julius Caesar of Rome by Morgaine la Fay. In Wolfram von Eschenbach’s “Parzival”, Arthur lineage is traced back to a knight named Mazadan whom a Fay named Terredelaschoye had taken to the magic land of Faymurgan. Terrdelaschoye seems to be Wolfram’s rendering of French “terre de la joye” meaning “land of the joy”, and it is possible that he has confused the named of the fay with the name of her kingdom, that is, in Wolfraim’s source the fay was named Faymurgan (Morgaine la Fay) and her land was the land of the joy. But it might also be that Wolfram’s source mentioned an unnamed Fay of the Land of the Joy who took Mazadan to the country of Morgaine la Fay.
One gets tired of those who want to explain the true medieval account of some feature in Arthurian tales, but who don’t know it themselves, and so present either modern inventions or even their own inventions.
But, generally speaking, the BBC series has very little relation to any medieval tales. Just as the writings of many modern authors, including Bernard Cornwall, have almost nothing to do with medieval tales, quite rightly, since they are trying in part to write about what might have happened historically centuries before the medieval tales. But mostly they are writing novels, intended to entertain, and they are making most of it up, because we don’t have any texts about Arthur that are generally agreed to be historical.
Nor is it generally agreed that Arthur was a “tribal leader”.
The BBC series is just as much an invention as the medieval tales and most modern novels, save that it follows medieval traditions even less than most of them. No reason it shouldn’t.