The main point is that lots of outstanding Shakespeare scholarship has been done based on the "William Shakespeare" paradigm. Re-evaluating that scholarship under the "Henry Neville" paradigm produces a much deeper understanding of that very scholarship.
I have found an astounding example of this in this paper from 1995:
Womersley, David. “France in Shakespeare's ‘Henry V.’” Renaissance Studies, vol. 9, no. 4, 1995, pp. 442–459. http://www.jstor.org/stable/24412297.
If we consider Henry V in the light of these general considerations, we can see that Shakespeare's depiction of the French is both complex and unusual. Shakespeare combined respect, and even compassion, for the French, with moments of scorn which were both more offensive, and aimed with greater precision, than anything in a precursor such as The Famous Victories of Henry V. Investigation of the high-political rumours current during the play's moment - the summer of 1599 - allows us to explain this distinctive enfolding of aggression within appeasement by reference to the likely interests of the Essexian faction the play seems designed to serve.Just read that paragraph and compare it with a profile of Sir Henry Neville. Read the whole paper and realize that the detailed knowledge Neville had, and was reflected in the play, was not actually current among everyone in London. It just appears that way because we now have access to people's private letters and government documents.
As ambassador to France, Henry Neville had privileged access to the latest information on every issue discussed in this article.
But more astounding is how Womersley's analysis lines up with Neville's political leanings. Aligned with the "Essexian faction"! What could possibly be MORE spot on? The discussion in the paper on the niceties of the "Salic Law" and its emphasis in the play... Just amazing. And he brings Richard III into the discussion... The most Nevillian of all Neville plays for a dozen different aspects.
Lots more to digest in this paper, but that's enough to make the point.
And this answers the question, of course, of why Neville wanted to publish anonymously. This way, he could express political opinions to a wide audience without attaching his name to it. Shakespeare gave him cover to express his views openly in a way he couldn't by writing plays as "anonymous".