I post a link on our fan page. Perhaps 10-20% of people who liked the page see it, due to Facebook algorithms.
I also share or post a copy to my personal page, and my wife's page. There perhaps 30% see if due to the interactivity factors of the Facebook algorithm. FB is getting greedy and wants you to pay $$ for other and friends to see your posts.
I have not tried FB groups, but other groups I am a part of (indie author, amazon self publishers), just get bombarded with self promotional links. Interactivity seems low. What appears popular is all the cat videos, weird top lists, celebrity stuff, all that.
Posting a video natively is an option I would consider when Facebook decides to share the revenue they will make from your videos. Sure you may gain exposure for your brand by double posting your videos to Facebook, but you have to consider how many people will click-over to YT from FB and if that traffic is worth the revenue that FB makes at your expense without sharing with you. Add to this the fact they want you to pay decent money for your friends to see your posts, I avoid FB.