This is hopefully going to be fixed in core soon, but if you’re running into this problem, the solution is to add the following functions to functions.php:

// Enbed YouTube Videos over SSL
wp_oembed_add_provider('#https://(www\.)?youtube.com/watch.*#i', 'http://youtube.com/oembed?scheme=https', true);
wp_oembed_add_provider('https://youtu.be/*', 'http://youtube.com/oembed?scheme=https', false );