The so-called 'Four Branches' have between them neither the same, nor indeed any fixed author. Nor would extant texts in the White Book of Rhydderch or the Red Book of Hergest be the original literary incarnations. Literary and linguistic dating is fairly esoteric and somewhat beyond my ability to intelligently comment upon, but Sioned Davies (I think in Four Branches of the Mabinogi (1993)) suggests the 11th century† as a probable initial period of oral and literary interfacing.

Interfacing since, as you suspected, these were not wholly literary creations, and incorporate and invoke medieval oral traditions which appear in both their content and composition. Pan-Celtic and Insular literary and oral traditions can be seen in the texts themselves, alongside other contemporaneous literary influences, notably from France which itself so readily seized upon Britain's Arthurian exports.

Although it significantly preoccupied (or even spawned) early scholarship of oral tradition, the approach of uncovering aboriginal forms of these traditions and narratives has been largely abandoned, for multiple reasons, though not the least of which is that it is often simply impossible to determine.

 

^(† In fact she provides the amusingly specific range of 1050–1120 in John Koch's Celtic Culture. I presume there is some linguistic or literary co-relation that lends itself to that window, but I can only offer it without further comment.)

Short answer: no, but debate continues. The strongest arguments tend to come from those who argue for a very late eleventh or even twelfth century origin.

Longer answer: It depends which translator is your favourite. I tend to side with "pan-Celticists" Patrick K. Ford and T.M. Charles-Edwards. The latter has written an excellent rebuttal to the argument for the thirteenth century advanced by Saunders Lewis and one for 1060 from Ifor Williams). {The entire article is fascinating and I'd highly recommend it to anyone with an interest in Middle Welsh philology.}

T.M. Charles-Edwards begins by addressing Ifor Williams' proposed date of 1060. Williams chose this date, he says, because of the political situation of the time. Williams claimed that there was a relatively small window of peace (under one powerful king) where travelling minstrels and poets would have been able to move easily around Wales. Given that the characters in the Mabinogi spend time roaming around the Northwest, south-west, and eastern portions of Wales, he believes that this would only have been possible if a single ruler were in charge to ensure peace and stability.

It's a rather specious argument as there are no instances of the opposite being true (that in times of war, poets stayed close to a single region). In fact, we know that in 1176 a lord in a western cantref allowed poets and musicians from neighbouring lands to attend an organised competition in his domain. As Charles-Edwards states:

The history of both Ireland and Wales shows how political disunity and cultural unity can march hand in hand.

Saunders Lewis' argument for a thirteenth century date relies on the presence (or absence) of various loanwords from primarily French origins and by making sweeping statements regarding the similarity of rulers in the Mabinogi to rulers in the real world (namely Henry II). The arguments against the latter are far too lengthy to get into here, so I'll limit my summation to Charles-Edwards' argument against the first point.

Lewis believes that the presence of various Middle Welsh words (teilygdawt, anryded, anwybot, and ansyberwyt) in the beginning passages of the First Branch "Pwyll" indicate hierarchical feudal custom, which he assumes were propagated throughout Wales during the thirteenth century. He provides no evidence for this claim and wrongly assumes that Celtic societies were rather more egalitarian. As Charles-Edwards says:

there is nothing un-Celtic about hierarchy and related rules

Later, Lewis claims that another word (iarll) definitively dates the Four Branches to the thirteenth century. Professor Kenneth H. Jackson (in Language and history in early Britain: a chronological survey of the Brittonic languages 1st to 12th c. A.D.), though, has said that iarll has appeared in ninth century literature.

I could go on and on, but the point is that Lewis' date should not be given any serious consideration.

Charles-Edwards is reluctant to give his own specific date, but has no problem accepting a range. From page 288 of the 1970 Transactions of the Honourable Society of the Cymmrodorion, he writes:

I do not think any precise date for the Four Branches can yet be given. The arguments of Mr. Saunders Lewis and, before him, Sir Ifor Williams, are not sufficiently convincing. [...] [t]he Four Branches cannot be earlier than the eleventh century.

Emphasis mine.

This isn't exactly true, though. The characters that appear in the Four Branches might in fact be based on archetypes that are far older.

In the brilliant preface to his own Mabinogi and Other Medieval Welsh Tales, P. K. Ford advances the idea that the Four Branches have inherited at least a bit of their character from earlier centuries. He is not the first author to do so. W. J. Gruffydd has stated that the story of Pryderi bears striking resemblance to the native British god Maponos, who was worshipped in Roman Britain. In the 1974 - 1975 Transactions of the Honourable Society of the Cymmrodorion, Professor Eric P. Hamp only slightly disagrees and claims that the Four Branches mimic the life of the father of Maponos. He writes the the word Mabinogi meant:

the (collective) material pertaining to the god Maponos

As /u/innisfallen has already talked about Sioned Davies' translation, I won't make much mention of it here, but will add that after reading her book, I had the impression that she believes them to be more medieval folklore than the possible remnant of earlier centuries.

The authors of the other two widely published translations didn't - so far as I know - voice an opinion on the putative origin. Lady Charlotte Guest was the first to publish an English translation in 1845, but she made a few errors that were corrected by the Gwyn Jones and Thomas Jones 1948 version (though the unfortunate misspelling of Mabinogion for Mabinogi remained).

Mods, I've never posted on your excellent subreddit before. If I need to add a separate list of sources, please let me know.