If you view source, you can see there are instances of <div> tags containing that copy. Eg on the Living page this shows up:
^ on line 2849
I would argue that json data should be in a script tag not a div.
Either edit the theme, test turning off features until those go away, or pay someone to sort. Then recrawl the page and see if it updates the Google index.
You can influence what pages Google selects organically by changing what pages have the most internal+external link equity, what is in the Top Level Nav and a few other things.
You can also remove by no-indexing a subpage, and recrawling and it will drop out from site-links. This will drop the visibility of the page in Google completely though, so usually will have some pretty major follow-on effects.
But directly changing them nah not a thing.
Good overview here: https://ahrefs.com/blog/sitelinks/