Full site editing/FSE: FSE template HTML not rendered

I adjusted the template_include filter to let the block-templates be loaded:

add_filter('template_include', function ($template_file) {
    global $_wp_current_template_hierarchy;
    $_wp_current_template_hierarchy = array_map('basename', $_wp_current_template_hierarchy);

    $views_dir_path             = resource_path('views');
    $template_file_base_path    = str_replace($views_dir_path, '', $template_file);

    $template_file_base_path_fi = pathinfo($template_file_base_path);
    $template_file_name_base    = str_replace('.blade', '', $template_file_base_path_fi['filename']);
    $template_file_base_path_fn = $template_file_name_base . '.' . 'html';

    $theme_path        = get_template_directory() . DIRECTORY_SEPARATOR . 'block-templates';
    $fse_template_path = $theme_path . DIRECTORY_SEPARATOR . $template_file_base_path_fn;

    return $fse_template_path;
}, 19);

However, the block template HTML simply appears without the placeholders being rendered.
Apparently these block-template files are passed through the blade-renderer and not through the Full Site Editing server-side frontend renderer.
What other hook do I have to adjust?