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...based on what I see in Google/Bing/Yahoo - it does seem to be a major factor.
There are also potential downsides to implementing sitewide redirects... roughly a 15% loss in PR to redirected pages, and lots of places to mess up.
The keyword in the URL isn't the main objective
Adding an extra level of folders is asking for trouble.
Quite often, particularly post-Panda, when I see people doing something "for SEO", it's because they've lost rankings and, as often as not, that's because their content is weak. That can also be reflected in their backlinks. In this case, it sounds like you're not ranking for "widgets".