This is true with the caveat that Holinshed put together the largest collection of works of others. Lear, eg, can be traced back to Historia Regum Britannia and 1136 and we out assume therefore that his sources were readily available texts for review without the Chronicle.
Here below are Wiki entry summarizing source texts -- I would like to EMPHASIZE that sources are therefore from the spoken or oral tradition and that 'first texts' are not creation dates for the stories.
It will no doubt annoy those who do not like William Shakespeare as author to learn that at Stratford he had a Welsh tutor, Price, himself a graduate from the rigorous memory and recital studies of that School.
This is worth a mention since sources for stories are always considered to be texts -- rather than from the oral tradition.
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Wiki records:--
The question of the dates of the tales in the Mabinogion is important, because if they can be shown to have been written before Geoffrey of Monmouth's Historia Regum Britanniae and the romances of Chrétien de Troyes, then some of the tales, especially those dealing with Arthur, would provide important evidence for the development of Arthurian legend. Regardless, their importance as records of early myth, legend, folklore, culture, and language of Wales is immense.
The stories of the Mabinogion appear in either or both of two medieval Welsh manuscripts, the White Book of Rhydderch or Llyfr Gwyn Rhydderch, written circa 1350, and the Red Book of Hergest or Llyfr Goch Hergest, written about 1382-1410, though texts or fragments of some of the tales have been preserved in earlier 13th century and later manuscripts. Scholars agree that the tales are older than the existing manuscripts, but disagree over just how much older. It is clear that the different texts included in the Mabinogion originated at different times. Debate has focused on the dating of the Four Branches of the Mabinogi. Sir Ifor Williams offered a date prior to 1100, based on linguistic and historical arguments, while later Saunders Lewis set forth a number of arguments for a date between 1170 and 1190; Thomas Charles-Edwards, in a paper published in 1970, discussed the strengths and weaknesses of both viewpoints, and while critical of the arguments of both scholars, noted that the language of the stories best fits the 11th century, although much more work is needed. More recently, Patrick Sims-Williams argued for a plausible range of about 1060 to 1200, which seems to be the current scholarly consensus.