Fixes #1792

Let's just source these samples directly from the generated files. That way, they'll always work even if the site is down.

Note: for formula.json, it's a little more tricky because the generated file is actually a template that is only expanded after the site is generated. Instead, just modify the formula/wget.json file to match the syntax that would be used in the larger file (indent, add surrounding brackets, and remove the analytics).

For the "docs" workflow, we currently generate the API samples to run the API docs page through HTMLProofer. As far as I can tell, this just checks for broken links and malformed content, so we can just put template contents in for the sake of this check. This avoids the hassle of actually generating the API files or importing them from a separate actions run when we don't need to.

And, while I'm here, let's add explicit contents: read permissions to the workflows that don't have them defined.