This is a screencap of Crosstalk after I had posted a tweet and added a video, plus a link. I kept the crosshead because I felt it enhanced the post internally. If there's not going to be way to keep them internally, but turn them off externally (increasing clicks), I think the video thumbnail should go over to the side with the crosshead, if it has to be visible outside the post.
Just an idea, but also one I've thought of before.
ETA (A bug report): If you run a Kinja search to see if anyone else has posted about something, such as I did before the one pictured or if you're on a tag page, you can compose a post to your heart's content, but Kinja won't let you publish it or save it to drafts. You have to point another tab elsewhere, then copy and paste your original post's content for it to save.
2nd Edit: Though the crosshead is showing in the WYSIWYG box, it's not coming up in the published post right now, for me. I believe that's happened at least once before that I've noticed.
TYLTK - Keep up the good work.