In blocks.wp-a2z the widget areas are mostly populated by shortcodes.
These don't appear to work in the template parts used in Full Site Editing.
Is this a limitation of Full Site Editing or am I doing something wrong?

Answer: Something wrong.
I added a shortcode to the content of the page and they all started expanding.
It looks like I'll need to implement some logic to ensure shortcodes are enabled when it's a FSE theme.
It could be as simple as do_action( "oik_add_shortcodes" ); in functions.php