@Green and @knigarian, hi! 
These are both really good points, which I think I can address together, because they come back to the same principle.
I made sure I wrote that an atta is an essence “even if theoretically not identified with”. Psychologically, however, the belief in some permanent entity in the person I think will always be accompanied by a sense of identity, even if very subtle. It’s not like such a permanent consciousness (or whatever permanent entity is proposed) would continue on without “me”. I’d be getting something out of it, namely some part of my being continues on. So although in theory one may pose an atta free from identity, in practice any such idea, when believed to be true, always goes together with some identification, with some sense of ‘I’.
This goes the other way too: if there is identification there is also a view of atta. This is one reason materialists are considered annihilationists. They still identify with something, be it the body or whatever, seeing it as the essence of their person, hence their self. This person or self they think ends at death, hence it is annihilationism.
The materialist scientists nowadays do not fundamentally different from the materialists at the time of the Buddha. In DN2 Ajita Kesakambala’s view is also explained without explicit mention of an atta. What he does is equate the person to the body which is made of the four elements, not to some “mental entity”.
By saying there is some real person-essence, he is still posing a self, as is clear from DN1, where the materialist view is described as annihilation.
Correct. Here we come back to the idea that the theory of a self psychologically can never be separated from a sense of self, even if theoretically it can.
The meaning of sakkāya is somewhat obscure, and the term hard to translate. Venerable Sujato discusses it here. I’m still processing his exact arguments, but on the whole I agree it is not just a sense of identity. (In fact, I pointed this out to him when commenting upon early translations.) Sakkāya is any kind of existence, apparently also if not identified with. Hence Sujato went from ‘identity’ to ‘substance’. If this is right, then it aligns with my interpretation that any kind of permanent essence (or substance) would be considered an atta. That would make any permanent consciousness also included in sakkāyadiṭṭhi. Sakkāyadiṭṭhi being the wrong view that things have a true essence/substance that lasts.
This reasoning feels far-fetched, and your interpretation is unique. This peculiar phrase on mother and father has been a matter of debate. We discussed it to some extent here. I don’t think it poses any real problems for either of our interpretations regarding nibbāna and consciousness.
We’re getting a bit off topic here, so for now I’ll leave it at this.