In Old English the most common single inflectional class are the masculine a-stems (counting the light and heavy subdeclensions of the neuter a-stems, feminine ō-stems, u-stems, and feminine root nouns as distinct), which form their plural in -as.
Reduction of final vowels would have led to almost total syncretism in the light feminine ō-stems (the next largest inflectional class), so analogy to some other class would be necessary, and we would expect this to (largely) be to the largest other inflectional class.
At this point, a large enough proportion of all nouns are following the old masculine a-stem inflection that it being used as a source of analogy for the remaining heavy ō-stems and neuter a-stems is natural and from there we arrive naturally at the situation of Middle English with two main inflectional classes: strong (with a plural in -es) and weak (with a plural in -en) with far more nouns in the former category. The development from that to Modern English is entirely expected.
At no point is it necessary to invoke influence from Norman French to explain the evolution, so by Occam's razor we should not accept this as evidence of Norman influence.
Note that Continental Germanic languages, which often do not use an -s plural so much started from a different point. In Old High German masculine a-stems formed their (nominative) plural in -ā (or -a). Likewise whilst Old Dutch does show masculine a-stems with a (nominative) plural in -as, they also show it in -a so it is no particular surprise that, like German, Dutch does not generally exhibit plurals in -s today.
The other Germanic languages that have lost similarly large amounts of their morphology are the (Continental) North Germanic languages, and there we see near universal use of a plural in -r, seemingly cognate with English's -s, from a dialectal difference in Proto-Germanic as to whether the masculine a-stem nominative plural ended in -z or -s (likely due to analogy between nominals with differing stress patterns prior to the application of Verner's Law) and of course there is no question of this being due to external influence in this instance.