The grammar is simpler (no gender and fewer inflections), but it has more phonemes than most other Indo-European languages, and has one of the largest vocabularies.
English is not a pidgin language because the grammar is very clearly identifiable as a Germanic language (albeit a clearly simplified form), but with substantial lexical borrowings from other languages. Grammatical simplification and lexical borrowing is not particularly unusual in language development--in fact, English may not even be the biggest offender in these departments (I believe Turkic languages borrowed more from Mongolian languages than English has from Romance languages).
Probably, I would say, the big reason for the lack of Celtic borrowings into English is the fact that, to whatever degree there was admixture between Celtic and Anglo-Saxon populations, English would have been the prestige language, and the prestige language tends to be little affected by borrowings compared to the non-prestige language. You see this with French borrowings: French was the prestige language of England for about 400 years or so, so English starts borrowing all sorts of things from French, but not vice versa. The existence of the Danelaw region--a part of England that was primarily Danish rather than English even when largely ruled by the English--provided a substantial contact region for Old English/Old Norse borrowing, which was helped by the very similar nature of the two languages.
The linguistic evidence has always suggested complete replacement. Even with the prestige language of invaders taking over you would expect some geographical names, especially for small, insignificant and / or out of the way features to persist and yet in England there is pretty much not a single place name that has Celtic origins (outside of Cornwall of course)