Are you thinking about a YouTube logo in the bottom-right corner of the embed? Were you expecting something like this?
If so, we've always had that disabled on here - when I was originally setting up the embeds, I figured the additional button was cluttering up the relatively small space.
As you pointed out, clicking on the video's title provides the same functionality. However, if people feel they'd rather have the dedicated button there, it can be very easily added.