SuttaCentral
There is a Dhammapada verse which has a nice way of putting it:
Dhammapada, chapter 5 verse 3:
"The fool worries, thinking, “I have sons, I have wealth.” Indeed, when he himself is not his own, whence are sons, whence is wealth?’
The bold in Pali :
Atta hi attano natthi ( he ,himself, is not his own)
There is no indication that atta is an illusion, only that it is not yours,it cannot be owned.
To restate my idea from earlier, I wonder whether the awareness of anatta could be accomplished by being aware that things are happening of their own accord rather than being under your conscious control. For example, say you are aware of some sensations as being temporary. Are the corresponding appearances and disappearances under your conscious control? If not, then maybe being aware of that is being aware of anatta.
Clip your toenails.
At least that one’s simple. 
Which bits of the process do you identify as being you? 
I don’t.
All of it! I still have a distinct sense of me and mine. Is it just me?
Imagine people would say ‘Yes, it’s just you, weirdo’ 
How can you observe something that is not there (anatta-sanna)?
Let’s also keep in mind that animals don’t have the cognitive-linguistic tools to ‘identify themselves’ with anything. So whatever the process is, it is pre-verbal, or meta-verbal, and still works flawlessly. Which means that our conscious efforts have to be accompanied by unconscious efforts as well. Hence, the ‘direct observation of anatta’ can’t be the end of it all.
You can directely obseve the sense of self. This means you cannot directly observe no-self, as that would mean it would need to have some sensation associated with it!
However it is possible to logically understand that there cannot be a self (ie to logically understand, without a doubt, that the mind and the body cannot be a Self)! This is falling away of Self View, a wrong View, rather than a bodily sensation!
The sensation of a self persists much like the flower petals still have the scent of a flower, long after the flower disintegrated, but there is no self View anymore.
Im not talking about an intellectual view, but about a deep sense of “me” as an individual biological organism.
Intellectually speaking, do you believe there is a self? Arriving at right view is the first step and wiping out the sense of self then becomes a possibility, and a desirable practice.
I believe I’m an individual biological organism, a thinking animal, and observing what arises at the sense-bases supports that belief.
So it’s a belief rooted in experience, not an intellectual view.
I’m afraid the lack of control argument doesn’t work for me. Clearly a lot of things are beyond my control, but that’s true for any living creature.
It feels like a argument based on an invalid assumption: “If there’s a you, then things should be in your control”. But why should they?
It’s a contrived argument, rather like the one which says the aggregates are not fit to be regarded as self.
A robot, which senses the enventment, monitoring that which arises at its senses could be also programmed to think ‘I sense, therefore I exist’. However this doesn’t mean the an ‘I’ exists, as the person doing the sensing, or apart from the sensing.
But objectively the robot does exist as an autonomous unit. You can programme it to make decisions, protect itself, and so on.
“I, Robot”. 
Yes, the robot’s parts and programming ‘exist’, if temporarily…
What role does self-awareness play here, particularly when enhanced by sati?
Could you elaborate on your reference to making “unconscious” efforts, as well as conscious ones?
Watching whatever you think of as the self, and seeing it’s impermanence moment to moment, or transience, which is ever present, shows that there is nothing that can be held on to, as a self, and watching impermanence also leads to full Cessation of the Five aggregates.
I don’t find that awareness of impermanence makes much difference to the sense of me as an individual biological organism, or thinking animal.
It makes a difference in terms of not attaching to it so much, a greater acceptance of ageing and eventually death.
But returning to the OP, I don’t think you can observe anatta directly. The methods we’ve been looking at involve challenging an assumption, though I don’t think the EBT are always clear about what that assumption is.