I think share-ability is important (seo/community, think of github repos). I'd like to do something like google for "precursor datepicker" view/fork or 'save'/star/insert it or maybe search in app for "datepicker"s.
That just my two cents, I think it creates a community and then as a whole everyone benefits, while the tool still remains uncluttered (the workflow remains the same, but components could be incorporated). Consider this: I could have say 4 tabs open which I then copy & paste some components I found into a mockup I'm working on. No searching in app, no component drawers, just simple copy & pasting between tabs.
Also, tracking a clipboard doesn't really classify something as "important", I mean, won't that list just get cluttered, I don't necessarily want to look through 20 copies to find the one time I copied the whole component. And workflow to update it might be a bit weird, do I edit it, re-copy it and re-star it, delete the previous staring?
Really, just my two cents, food for thought, great work, I'm sure you know better!