Again this is my opinion.
Very few classes have anything innate. The Sorcerer class, to me, seem to be the only one that should be considered in anyway innate. ASI is TRAINING as in lifting weights, etc.
The Lefty build using the Rogue class and classes feature expertise so claiming that his investigation skill is innate doesn't track.
That's a reasonable opinion, but as you asked about my beliefs about the matter, and my belief is: it depends on your definition of "innate".
For example, a
possible definition is that aspects of the character that are present
ab ovo when the character is created, and cannot be modified by experience (levelling up) or downtime training, are innate. Anything that is fluid and develops in-play is not innate.
If we follow that definition, abilities are not innate, skills are innate (mostly, but bards can add more), racial traits and background features are innate, and class features that are assigned at level one (or higher, if you create a higher-level character from scratch) are innate but class features acquired subsequently are not.
So that definition of innate seems to lead to the conclusion that class features are partly innate and partly not.
But that was just an example.
You see, I don't have a
single belief about the answer to your question. I don't believe 'yes' and I don't believe 'no', I believe 'it depends on your definition' and I also believe that a great many different definitions are tenable. There is a whole universe of possible answers and I "believe" all of them.
But does it matter? Does it help to distinguish innate class features from non-innate ones? And even if it does help you, does it really matter to you what BoldItalic believes about the question? Before I can give you a more helpful answer, I need to know why you asked the question.