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It looks like Moz's SERP analysis Reports are almost entirely based on link analysis. Does that mean that this isn't very relevant in my area?

I think the API will help, but for the same date range, no filters, etc, the data shouldn't have changed. BUT Google has been known to edit their Search Console data, or they have in the past when they found discrepancies. There are any number of reasons why, but I am sorry we couldn't nail it down for you. I really do think the API will help. Best of luck!