Here is my question: Is there a way to have YouTube host the videos, but only be able to see them embedded on our website? That way people would only be able to link to our website if they wanted to show the video. Dumb?
not dumb, but there's no technical way around this. By taking advantage of Youtube and using their enormous reach network as well as using as much bandwidth as you please, you also give away the rights to your video in some ways.
If your'e not going to host the video yourself then you need to have other reasons why people would want to come and visit your site.
Is the video the primary piece of content? Or could you offer an article that adds more substance to the video? By adding the additional content you not only make the page that is embedding the video more worthwhile to link to, but you're also increasing the content on your site and improve the overall SEO and, most importantly, improve the users experience on your site.