How to print a taxonomy term's field value alongside a Views grouping header?
Thanks to Mario's helpful hint in the first comment, I was able to do this in the View and not in the twig template:


In the FIELDS section of the View, add "Most Common Name" as a field.
Select "Exclude from display", because I only want it in the title, not the results themselves.
(3. Select "Link to the Taxonomy term", because I want it to be a link.)
Next to "FIELDS", under the "Add" drop-down menu, choose "Rearrange".
Drag "(field_primary_taxon: Taxonomy term) Taxonomy term: Most Common Name" above "Media: Primary Taxon" > Apply
Click "Media: Primary Taxon [hidden]" field
Check "Override the output of this field with custom text"
REWRITE RESULTS > and copy the required tokens from the "REPLACEMENT PATTERNS" list, and paste into the "Text" area. I also inserted brackets outside of those tokens, so my Text area looks like this:

{{ field_primary_taxon }}
(
{{ field_most_common_name }}
)
Apply > Save > Clear all caches if necessary > Refresh the page.
Title now in my example is "Animalia (Animal)". "Animalia" and "Animal" are links to the taxonomy page.