I've always carried the pet theory that it would be good to translate "anattā" as "identity-less" or "lacking identity", but I may be alone in preferring that rather strange rendering.
No self or not self discourse, whichever one prefers, I think ultimately and practically deals with the discarding of self-views. Even if you are of the persuasion that perhaps some sort of true self/ātman does exist, and that the Buddha teaches the way to it, even then one would still have to deal with all dhammā lacking this essence that one posits as a potentiality, so all self views still have to be discarded for what they are, views.
With this in mind, I personally think "without identity" is a more natural coinage for the English language and English language culture, although I understand that this suggestion is idiosyncratic, and I would certainly never insist upon it being "proper" in the sense that I would suggest others use it. "Self", in English, has a natural metaphysical weight that makes the negation of self seem like a negation of kamma almost, a negation of responsibility, moral responsibility, morality, nihilism. Such is the "Self" in English, in my experience at least.
Identity we are used to thinking of as fluid, as unstable, as unsatisfactory. In this modern day age our "identities" can be stolen, appropriated. Identities are fictionalized and digitalized on the internet. English speakers very naturally associate identity with incoherence and fiction.
Futhermore, identity has a broader, more substantial meaning, that being, any way in which one considers or regards one's "self" or anything other than something conventionally considered a "self", to be a "self". Identity is necessarily formed via identification of self-qualities, or self-parts. "I am an X. I am a Y." Identity in English, in many ways, is similar to the Buddhist Hybrid English of "self-view". There is even another interesting Buddhist Hybrid English term I have encountered in use before "I-making" or "self-making", both of these are very readily relatable to "identity", as a term, to the English-language neophyte, and once all self-view is understood to be (merely?) "identity", one has the makings of proper instruction on what is expressed as "is not the self" in traditional Buddhist language. I think, at least.
In the modern era, we even have meta-identities, or hypostatizing identities with identifications within them, how often do we see "I am offended on behalf of X identity because I am Y identity and X and Y identity are united against Z identity"-type discourses abounding in all spectrums and manner of society, obfuscating real and pressing issues that underly these identity warfares, in the present age?
It is not an EBT, but I am reminded of the opening dedication of the Mūlamadhyamakakārikā, "prapañcopaśamaṃ śivam", " the auspicious cessation of reification", I think the cessation of "identity-forming" and "identification" (both of those being, I think at least, forms of "reification/objectification/substantialization/hypostatization") is an important part of the Buddha's teaching, but that language was not really used by the Buddha, so at best it can only really be my own personal responce to anattā, be it on or off track.
Anyways I hope that someone else finds something interesting in those sentiments.