As things currently stand, a CSV export from iNat will never include the values from Annotations. This is mainly due to the exporter being quite old, and Annotations being relatively new.
One alternative is to use the Explore page to filter out only observations that have a certain annotation, and download that set. It’s not a great alternative, but it’s doable. You would have to edit the search’s URL, as shown here.
For example, here is a search for all Asteraceae in Alameda County (where I live), that have been annotated as flowering: https://www.inaturalist.org/observations?place_id=845&subview=grid&taxon_id=47604&term_id=12&term_value_id=13
That last part, &term_id=12&term_value_id=13, determines which annotation and value(s) are included. Then I can export this set of observations, knowing they have all been marked as flowering.