A 301 assumes that the old site is obsolete and no longer required, which I think you said is correct?
The 301, in this case, will assist visitors following old links, but there will be minimal "SEO" benefit, as the new site is quite different, so will be found by different search terms. For a week or three, while the old site lingers in the serps, there will be some benefit.
There is no way to effectively transfer SEO benefits where the content is utterly different, as we've explained already.
But if the old site did well in Google, those old links must be doing something right, so the 301 will forward visitors, until the 'linkers' realise the site has gone and remove the link, anyway. but that could be a while.
What you have to do now, is find a way of keeping 'old widget' fans on a 'new widget' site, once they arrive!
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