Do you know a video plugin that allows embedding in the home.php file?

The excerpt is not meant for content that is why it is called “the excerpt”.

Instead use a plugin like “Vipers Video Quicktags” or “Youtube shortcode” and add it to a template file and you’ll probably want to get the url via a post meta field.

for example ( using Youtube shortcode plugin and meta field called “video”):

$video = get_post_meta($post->ID, 'video', true);
echo do_shortcode('[youtube_sc url="$video"]');