We are just using the embed code from YouTube to embed the video in our own web page. I'm not sure if there is anything we can do to improve it.
Do you have the same experience with videos directly on the YouTube site? Does the screen reader see the needed buttons on the videos on YouTube site? If it does work better directly on YouTube then there may be something we can do to improve it, but if it works the same there then probably not.
Best,
Joe