How is the Anatta (not-self) doctrine useful?

It is "useful" like you said. But I would like to mention that in the context of society, it is useful to assume an ego. Without it, we would not be able to function responsibly. But of course, be aware that you are just assuming it and responsibly commit actions.

It is a very interesting question @Arthur A. Jacomelli, that you have raised. There’s a famous simile in the texts where a man is suffering from pain. He gets tied up in anxiety and misery around the pain. And the Buddha says it’s like being shot with an arrow and then shooting yourself with a second arrow. The physical pain is the first arrow; the mental pain is the second one. And it’s the mental one that’s important. As the Buddha says, the enlightened person, the awakened person, may still get shot with those first arrows but doesn’t shoot him or herself with the second. It is because an Arahant has gone beyond the last of the four kinds of clingings – the Attavada Upadana (attachment or upadana is “firmly grasping”.) Attavada is the last of the attachments which is in other words a hindrance to the realization of Nibbana.

So our training is learning how not to shoot ourselves with that second arrow. The question is: How do we learn not to do that? If a person who has gained very exalted states in meditation and responds by saying, “I am at peace. I am released”, the “I am” in those statements is what’s causing the problem. It shows that this person still has some connection, still has some clinging. After all, craving combined with clinging is what causes the suffering. You impose that idea of who you are on all kinds of experiences.

This is where the teaching on not-self comes in to help. Learn how to view things without creating that sense of self—because after all, that sense of self is something that we do. We make this sense of self. And it does have its functions. As the Buddha said, when you want to understand something, you have to see both its allure and its drawbacks. You don’t just watch it arising and passing away. You want to understand when it arises why you hold onto it? So if you want to understand why you make a sense of self, you have to look for its allure. And the allure here is that the sense of self is useful in a lot of contexts. When you’re eating food, you know which mouth to put it in.

It’s precisely at this fork in the road where the analysis of sabbe Dhamma anatta—all dhammas are not‐self—applies: where you might see nibbana as a dhamma, as an object of the mind. As long as you perceive it in that way, there’s going to be attachment, there’s going to be a dhamma to hold on to. So you have to learn how to overcome that attachment by applying the perception of not‐self to the dhamma of the deathless. Then, the texts say, you let go of all dhammas, which allows you to see nibbana in another way—not as a dhamma, but as the abandoning of all dhammas. That’s the ultimate. And at that point, these three perceptions lose their function. They’ve done their work, so you can put them aside. After all, they’re conditioned phenomena. When you’ve put all dhammas aside, you put them aside, too. Arahants can continue using these perceptions as a pleasant abiding for the mind, to remind them of why they’ve got the ultimate happiness, but these perceptions are no longer needed in the task of bringing about release.

The 'self' is an idea/assumption born from craving for & attachment to things (Parileyyaka Sutta, paragraphs 9 to 11). When the things attached to pass away (due to impermanence) there is suffering because of the 'self-clinging' or 'self-identity' with those impermanent things produces the sense of 'loss'. The Nakulapita Sutta explains this very well & simply.

Therefore, yes. The purpose of anatta is to produce not-clinging to end suffering & reach Nibbana.

The Anatta-lakkhana Sutta answers your question, particularly the last paragraphs (below):

...all...must be regarded with proper wisdom, according to reality, thus: 'This is not mine, this I am not, this is not my self.'

O monks, the well-instructed noble disciple, seeing thus, gets wearied of form, gets wearied of feeling, gets wearied of perception, gets wearied of mental formations, gets wearied of consciousness. Being wearied he becomes passion-free. In his freedom from passion, he is emancipated. Being emancipated, there is the knowledge that he is emancipated. He knows: 'birth [of self] is exhausted, lived is the holy life, what had to be done is done, there is nothing more of this [self] becoming.'

This the Blessed One said. Pleased, the group of five monks were delighted with the exposition of the Blessed One; moreover, as this exposition was being spoken, the minds of the group of five monks were freed of defilements, without attachment.

I think the use is described in Alagaddupama Sutta,

Abandoning Possessions & Views

"Monks, you would do well to possess that possession, the possession of which would be constant, permanent, eternal, not subject to change, that would stay just like that for an eternity. But do you see that possession, the possession of which would be constant, permanent, eternal, not subject to change, that would stay just like that for an eternity?"

"No, lord."

"Very good, monks. I, too, do not envision a possession, the possession of which would be constant, permanent, eternal, not subject to change, that would stay just like that for an eternity.

"Monks, you would do well to cling to that clinging to a doctrine of self, clinging to which there would not arise sorrow, lamentation, pain, grief, & despair. But do you see a clinging to a doctrine of self, clinging to which there would not arise sorrow, lamentation, pain, grief, & despair?"

"No, lord."

"Very good, monks. I, too, do not envision a clinging to a doctrine of self, clinging to which there would not arise sorrow, lamentation, pain, grief, & despair.

The problem is that clinging to any doctrine of self would cause sorrow and so on to arise. So it's better to have no doctrine of, to not have a doctrine of, self.

I read this essay No-self or Not-self? as saying that the question of self ("Self or no self?") could lead to a doctrine of self if it were answered, and shouldn't be answered and/or an answer shouldn't be inferred (because doctrines of self lead to suffering).

The Anatta-lakkhana Sutta says that what's impermanent is unsatisfactory and that it isn't proper to regard that as self,

Now, what is impermanent, is that unsatisfactory or satisfactory?"

"Unsatisfactory, O Lord."

"Now, what is impermanent, unsatisfactory, subject to change, is it proper to regard it as: 'This is mine, this I am, this is my self'?"

"Indeed, not that, O Lord."

"Therefore, surely, O monks, whatever form, past, future or present, internal or external, coarse or fine, low or lofty, far or near, all that form must be regarded with proper wisdom, according to reality, thus: 'This is not mine, this I am not, this is not my self.'

In a sutta like Aggi-Vacchagotta Sutta he holds neither that body and soul are same, nor that body and soul are different; partly because,

the position ... is a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. It is accompanied by suffering, distress, despair, & fever, and it does not lead to disenchantment, dispassion, cessation; to calm, direct knowledge, full Awakening, Unbinding

The Simsapa Sutta suggests that what's being taught is suffering and the cessation of suffering.

IMO the Anatta doctrine should be understood as being in contrast to the Ātman (Hinduism) doctrine.

I think that at least two of the fetters are associated with a sense of self: "identity view" and "conceit" (see e.g. these answers about identity view and these answers about conceit for further details).

Another famous, relevent summary is at the end of the The Anatta-lakkhana Sutta:

Therefore, surely, O monks, whatever [of the five khandhas], past, future or present, internal or external, coarse or fine, low or lofty, far or near, all that form must be regarded with proper wisdom, according to reality, thus: 'This is not mine, this I am not, this is not my self.'

That "This is not mine" could be viewed as an antidote to greed (thirst or attachment which, according to the second noble truth, causes suffering): either an antidote like a medicine taken to cure a disease, or perhaps a description of the healthy state when the disease doesn't exist.

Similarly the view that "I am not this" might be an antidote to pride.

Finally, "this is not my self" (or "not my soul") might be the logical consequence of the previous two, i.e. perhaps it's what you'd believe (i.e. what your view would be) if you had no craving (desire for "mine"), and also no conceit or pride.

Moving on (from the Pali suttas to Mahayana-influenced doctrine), the Dalai Lama said that he views himself as "not special" and "just like other people": that otherwise (e.g. if he thought of himself as "the Dalai Lama" or as "the Nobel Peace Prize Winner") he would feel imprisoned.