Do I take this to mean that even an organization account cannot hold more than 1000 features per layer?
No, hosted feature layers can contain many more than 1000 features. 1000 features is a rough guideline for drawing the layer on the map, as the features are drawn as graphic elements by the browser, mobile app, etc. You just need to appropriately set scale dependencies. For example, you probably wouldn't want to draw parcels if your looking at a map of the whole country.
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