@Jennifer Loudiana - The embed code for a video on Stream (on SP) doesn't change ever for a video no matter what the permissions are.
The embed code works off of "direct permissions" meaning you have to have granted access to people via "grant access" or they've already been granted access through a SP site, folder, or a share link they already clicked. The users who will be able to see the video in the embed code can be found by clicking on "Manage access" which shows all the users who have permissions.
I'm not quite sure why you were seeing the blank spot, I've not heard of that before nor does it make sense if all you were changing is the permission on the video.
The embed URL does matter based on the site the file is stored in. If you created an embed code, then moved the video file to a different SP site or from ODB to SP, etc. then I that would break the embed code link.
If my above explanation doesn't help and you still have the problem then I would suggest working with an admin to open a support ticket so we can investigate more.