If you don't think widgets.com/red/ is more likely to be about red widgets than shshts.com/tsref/ then that's your choice. But I'm going to consider that an extremely bad one.
Grow up. It think that widgets.com/red/ that doesn't mention red widgets is less important than shshts.com/tsref that does. And if both were to mention red widgets then it would turn to anchor text, themes and so on.
These days the chances are the widgets.com/red/ is an affiliate spam site pointing to shshts.com/tsref/
And what does a company have to do with it? We are talking about websites. If your search engine doesn't seek to rank websites, that speaks for itself.
You do realise how ridiculous that sounds don't you?