This is a facebook question, not a TubeBuddy question, so I've moved this question to the social media section (Which is still new) I don't use facebook so I can't answer this, but this is a better part of the forum to ask.
The truth is that, Facebook suppresses YT videos posted on that platform. Facebook wants you to post videos natively on Facebook. If you post a link to a YT video, Facebook's algorithm will shadowban it so that it is not shown to people, while letting you still see it so you do not know that it is shadowbanned. There are ways around this, but Fb generally does everything they can to keep people from seeing YT links. A YT link takes people off of the FB platform, and FB does not want this because they make money from showing ads. Facebook has their own advertising platform and are not part of Google ads.
Facebook is a Pay to Play platform. If you want to promote something on Facebook, they want you to pay for advertising. Facebook wants you to give them your content so they can make money from it by posting ads on it, and then they also want you to pay for them to promote your content, so they can make more money off of it.
Autoplay on FB & YT is a feature that is controlled by the user. I set mine to not autoplay. Any site that uses autoplay gets turned off. I decide if I am going to play a video or not. Same with autoplay music. Nothing will get to bail on a page like autoplay media of any kind.
Visit the YouTube page of a video you would like to embed (link in Resources). Click “Pause” if you don”t want to watch the video while embedding it on Facebook.
Click “Share” to view the video”s sharing options. A list of social media logos are displayed next to the video”s page link. Click the “Facebook” logo to embed the video on your Facebook wall. You are prompted to sign in to your account and confirm your choice. Click “Share” to embed the video player on your wall.