FSE: oembeds do not work in block template parts

[johnstonphilip]

Description

If you put a youtube video (or any oembed) into a block template part, it doesn't get embedded on the front end.

Step-by-step reproduction instructions

1. Add a youtube video block to a block template part (like header.html)
2. Add that block template part to index.html
3. View the frontend, and notice the video was not embedded, but shows the youtube video URL only

Expected behaviour

The youtube video gets automatically embedded, like it does in a blog post.

Actual behaviour

The youtube video does not get embedded, but simply has the URL placed on the frontend in its place.

Screenshots or screen recording (optional)

WordPress information

• WordPress version: 5.7.2
• Gutenberg version: <10.7.1 |
• Are all plugins except Gutenberg deactivated? Yes
• Are you using a default theme (e.g. Twenty Twenty-One)? No ( I am using a custom FSE theme)

[johnstonphilip]

The reason this happens is because template parts do not get filtered through the_content like normal posts do.

See this PR where block template parts are no longer filtered through the_content:
#20343

The problem is that oembeds are filtered to the_content in WordPress core here:
https://github.com/WordPress/WordPress/blob/270f2011f8ec7265c3f4ddce39c77ef5b496ed1c/wp-includes/class-wp-embed.php#L39

So the fix will likely need be to run $wp_embed->autoembed on block template parts, like what is done for widgets in core here:
https://github.com/WordPress/WordPress/blob/1314542c50e2ad2075b1304df20471febffb8fad/wp-includes/widgets/class-wp-widget-block.php#L77