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Long story short, this is just a basic out of the box connector, not a direct integration with Google Analytics. We have almost no control over how it works. So, I would take this data with a grain of salt since it is not something we built and infromation around the connector is fairly vague.
The Natural Search part is actually controlled by the vendor (ie Google):
Account Engagement looks at the type of source (natural search, ad, paid search, or empty), and the vendor for the prospect’s first touch with your marketing assets. The two parts (vendor + type) are then combined to give you a value for the source field. If the type of source is empty, the vendor populates the source field.
For example, the vendor Google can provide all three types: Google Paid Search (your Google Ads), Google Ad (content network ads), and Google Natural Search.
This is all I could find on how that field is filled out. Hence, I would be hesitant to use it as a source of truth since we cannot see or change it’s behavior.