Some CSS classes missing when rendering the page with get_the_content()

You can achieve that by applying the the_content filters, just like what the_content() does:

$content = get_the_content();
$content = apply_filters( 'the_content', $content );
$content = str_replace( ']]>', ']]>', $content );

echo $content;

Or the other way is by applying do_blocks() on the content:

$content = get_the_content();
$content = do_blocks( $content );

echo $content;