Have you tried simply pasting the straight url into a post or page in the editor (don’t have to be in html mode)? YouTube (and a host of other services) should fall under oEmbed standard and render the video embed perfectly.
esmi is also correct as well that there are settings in the Admin Panel, found under Settings and then under Media. Once in the Media section, make sure you have “When possible, embed the media content from a URL directly onto the page. For example: links to Flickr and YouTube.” checked.
It can be found here: https://yoursite.com/wp-admin/options-media.php