"I ask this since I read that a best practice is to test only the public methods of a class. In this case I'm not testing public methods of a class but the public API functions from the whole component.
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I can't see the hair you're splitting. Public API is the public API.
Public API methods of a class or Public API functions from a component is the same thing -- a Public API -- the only thing you should test.
You have two levels of public API: class level and component level. That says you have two levels of unit testing. Class-level unit tests and component level unit tests.
Some folks will quibble on the definition of "unit" for unit testing. Some will claim that "unit" is always a stand-alone class and nothing more. I can't see how this is true, but some folks will claim it.
A unit is atomic or indivisible. It's a unit when your test does not reflect the unit's structure.