It’s seems that there are lots of things mixed up.
When you talk about Shopify API Feed, I guess you meant the google and youtube app, this app is to submit data to Google Merchant Center.
However, you are also talking about Google Search Console, which has nothing to do with the google and youtube app, nor the shopify api feed.
Google Search Console uses sitemap related data.
From the sitemap, it will crawl the pages, and use the info supplied on the pages. And more importantly structured data.
Structured data is submitted via the theme if you have coding, or in some cases an app.
If you are talking about Google Merchant Center, then 3rd party shipping data, can not be submitted to Google Merchant Center. If you are using native shopping options, then this should work, if it doesn’t update, you need to contact Google: https://support.google.com/merchants/contact/shopify_3p_support_app
If you can not fix the issue in Google Merchant Center, then use a different app such as: https://apps.shopify.com/multiple-google-shopping-feeds
We have skipped around in different directions, if this does not help you, then wait till someone else replies, or hire someone to look into your account to identify the issue and concerns you have.
I have tried my best to understand your issues, but I feel like it’s not something I can help with via community replies, as you screenshots and first reply is about Google Merchant Center, then you say Google Search Console. So it’s going all over the place.