Is there a benefit to using an embedded Youtube video versus AWS hosted video on a site?

Hello all. I'm hoping someone may know what's best in this situation. I have a site which has multiple youtube videos embedded on it. These videos on youtube are responsible for sending a significant amount of traffic to the site. Not all traffic comes from youtube though, so anyone who lands on the site through search engines can see the videos there, and they are important for visitors to watch. However, with them being embedded on the site, there are a couple of things that I do not like at all.

First, when someone is on the site and sees the videos, there is a link in the corner of the video that says "Watch on Youtube", and then at the end of each video, it will pull up like 12 related videos that will display on the video screen. I do not like either of those options, because they could both lead to people clicking somewhere and leaving the site.

So, it occurred to me that maybe it would be wise to use a video player, and host these same videos on Amazon Web Services. If I were to do that, there should still never be a problem with the videos loading, and it would not be a burden on my site server. This would also get rid of the "Watch on Youtube" link and related videos at the end.

Of course these same videos would still be hosted on youtube as well, and could still drive traffic to the site based on their keyword rankings there; they would just be hosted elsewhere on the site itself. Can anyone think of any reasons this would be a bad idea to host the videos on AWS versus using embedded youtube videos on the site? Is there a benefit to using the embedded youtube videos on the site that I'm not thinking of?

Thanks,