They are downplayed because the influence of Brittonic languages on English was pretty insignificant. There are vanishingly few Brittonic loanwords in English. Most of the few loanwords are from after 1500, well after the Anglo-Saxon and Norman invasions. "Paw" is from Welsh, which is just about the most common loanword. ("Dad" might also be from or influenced by the Welsh tad.)

Compare this to Latin, from which we derive an enormous vocabulary largely of technical and literary terms, and to French, which has blended into our vocabulary of everyday terms like place, chair, beauty, real, strange, and many other words so integral to English that you would not even suspect they were loanwords to begin with. In a typical dictionary 2/3 of the words will be from one of these two languages (though it is worth noting that the most common words are almost all from Old English).

Old Norse is somewhere in between. Its influence on English has been somewhat overstated (partly due to influence of the Danish linguist Otto Jespersen) but there is no doubt we derive a lot of common vocabulary from Old Norse, including the pronoun "they", plus words like "egg" and "shirt." There are also some scholars who contend that the centuries-long relationship between English and Norse settlers led to a simplification of English inflectional morphology (e.g. verb conjugations and noun cases) as the Norse failed to picked up the complexities of the language.

I should mention one theory, advanced by McWhorter in "Our Magnificent Bastard Tongue", that Brittonic speakers learning to speak English did import two important grammatical features from their native language - "do support" (starting questions with "do", as in "Do you like cheese?" instead of "Like you cheese?") and the progressive aspect ("I am walking" instead of "I walk"). When language learners use syntax from their native language in the language they are learning it is called a "substrate influence." If this theory is correct, it would be by far the biggest influence on English by Brittonic. But the argument has not gained widespread acceptance at present.