Turns out the thread I shared on your Facebook post was pretty close, you just need to append the name of the category to the end of _yoast_wpseo_primary_. So for the built-in categry taxonomy, you could display the term ID using <Field _yoast_wpseo_primary_category /> on a post where you’ve set a primary category. You could then pull that term ID into a taxonomy term loop if you wanted to display the term’s title.
Does this seem right? It doesn’t show anything so I’m sure I’ve done something wrong here. <Field _yoast_wpseo_primary_project_type /> does produce a number