You should absolutely upload your videos directly to Facebook. While YouTube is still king of online video, Facebook is rising quickly. In fact, Facebook now gets more daily video views than YouTube (over 3 billion a day).
So why not just paste a link to a YouTube video on your Facebook page? Well, only videos directly uploaded to Facebook will auto play in people’s newsfeeds.
How many times have you been on Facebook looking for something and spent 10 minutes watching a video of a cat shooting laserbeams out of her eyeballs or something? All because the video was already playing! The less barriers between your content and your audience, the higher your engagement.
Also, Facebook’s newsfeed algorithm gives native videos a lot of weight. Since Google and Facebook are competitors, they’re obviously going to rank their own video content over YouTube, and I don’t see this changing anytime soon.
The only exception to this could be longer-form videos or content that’s behind a paywall. Then I’d suggest taking little snippets or previews of the video(s) and sharing those directly on Facebook instead.