Hi Will,
The data on the paid traffic/ads side (product feeds, reviews feeds) are separate than on the organic side (page content, structured data).
The only crossover is that Merchant Center (paid side) will check your landing pages (organic side) to make sure they are legit and match the data from the data feed. One part of that matching is checking the structured data for the product price, availability, url, and product identifiers (sku, mpn, gtin).
Adding structured data in general does not change rankings, for better or worse. It’s more about improving how your existing search results appear by getting search enhancements that help your visibility, which can lead to more traffic and conversions.
I have a couple of articles that dig into this if you’re curious.
How to increase search traffic to your Shopify store using the rankings you already have
https://www.ilanadavis.com/articles/increase-search-traffic-shopify-store-using-rankings-already
How much Rich Results will enhance your search results
https://www.ilanadavis.com/articles/much-rich-snippets-will-enhance-search-results
It could be that your theme is missing data needed to qualify for Rich Results. If you share your link or email me at support@ilanadavis.com, I’d be happy to do a free structured data audit to see if JSON-LD for SEO could help you.