The "idea of self" is also the making of its existence. And we will see below that this existence is not totally its own.
There are indeed two concepts in the "idea of self".
The self must first be understood from the point of view of the pervasive and continuous Universal Self (with a big S); but also, from the point of view of being the result of a cause from nāmarūpa.
It is a self; in the sense that the "idea of self" has made it a real self; but it is not also "my" self. Therefore it can't be equated with the "big" Self. There is no entire pervasiveness and continuity in that self; because it is not "mine".
MN 22 explains the self pretty well. There is no self of his own (attaniya). Niya means "one's own".
What is said there is that, not only do we not have our own self, because we are the consequences of a cause that is outside of us; but moreover, we can't pretend to have that pervasiveness and continuity of the "big" Self, because of that.
Indeed atta should also be translated as "universal" and not just "self".
There is nothing Universal (big U) in the "self". And there is no "own-self" either.
The world (loko-SN 35.82) is not the Self/self; and the self is not our own.
But the self does exist. And this is the problem.
Because of the "idea of self", the self exists. It is not a self that belong to us. And it is not also a self that is a part of a whole.
Every time we see something, we appropriate it as the "I" (universal) and we make it "mine" (our own). 
SN 22.47 shows clearly these two particular courses of action. First we have the idea of a part that belongs to a whole, and is a whole in itself (the "I"). And secondly, because of that, we make the khandhas of nāmarūpa ours.
Moreover we mix both perceptions as a sole experience. We don't know how to discriminate through pañña, between what comes and belong from nāmarūpa, and what we have “built” as a self "proper"; as a license from nāmarūpa, so to speak; as a franchisee of nāmarūpa.

In other words, what is "mine" also exists paradoxically.
All this is the result of the wrong view of self. The "idea of self" creates the "I" and the "Mine".

The escape is paradoxically there. In the existence of that self, and in the possibility to make it "mine". If that possibility was not to exist, we could not differentiate between our mind and the rest (feeling > perception > intention of nāmarūpa). The cure is in the sickness. The cure is in the symptom. Before all, the cure is in discerning what is sick and what is sound in us; what is sick & sound in the inside.
The riddance of the nutriments is just the riddance of this "mine" part through the riddance of the sick part. This is sila (virtue).
We must get rid of what we crave and cling to. The result of this craving and clinging built along the recursive process.

When you stop smoking, the nutriment of sensual desire fades away, as your intention (cetana) to quit is fed to viññāṇadhātu; that descends in the cetana khandha of nāmarūpa; fed again to you, satta.
This is the recursive process of SN 12.65 and the establishing of consciousness.
What I mean by that is that the nutriment of sensual desire occurs in proportion of your own cetana.
Again it is paradoxical that something bad (the self) can also be the way to an escape.  But it is also the truth in the process.
Only the self can build akusala, and turn it into sila later on. But at the same time, (ultimately,) we have to get rid of the self and sila.

Notice that nāmarūpa is not just feeding you back with your intention proper; but also with its own intentions.
These intentions are far from being clear; but you can infer on that, with much certainty.

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The idea of self is always present and insidious.
For a young tender infant lying prone does not even have the notion ‘identity,’ so how could identity view arise in him? Yet the underlying tendency to identity view lies within him.
MN 64

The idea of self is an underlying tendency. Something that pervades our life from birth to death. It is latent. Until we uproot it thoroughly.
Vedic Brahmins did wrongly awaken to it. Some others have the Universal as religion; or they are just unconscious of it. There is this idea of a continuous self; even when we think it is not continuous. This is the continuity of delusion.
“Rāhula, whatever form in the past, in the future or at present, internal or external, hard or fine, unexalted or exalted, far or near, all that matter is not mine, I am not in it and it is not my self. Thus it should be seen according to what have become with right discernment (yathābhūtaṃ sammappaññāya passati ).”

“Rāhula, to him who discerns and realises in this manner, the latent tendencies of measuring in the form of "I be" in this sixfold conscious body and all external signs do not occur.”
Evaṃ kho, rāhula, jānato evaṃ passato imasmiñca saviññāṇake kāye bahiddhā ca sabbanimittesu ahaṅkāramamaṅkāramānānusayā na hontī”ti.
This latent view is what makes the "I" and the "mine". SN 22.47 is not the only sutta that states the twofold course of action; namely "I" > "Mine". There are dozen of suttas with parallels relating to that "I" > "Mine" making. But the mother of this process is wrong view about the self.
The cure is to see that the clinging khandhas are not ours. But at the same time, you have to "use" these clinging khandhas to differentiate. The right perception can only come from the realisation of the wrong perception. But the wrong perception must be there. You have to get rid of something that exists.
Self is a paradox.
It is of the same nature than the paradox of craving in AN 4.159 (SA 564), where it says:
"This body comes into being through craving. And yet it is by relying on craving that craving is to be abandoned."
It is a paradox.
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Attakārī Sutta: The Self-Doer (AN 6.38/SA 459) is a great reference.
Piya Tan's Āgama Nikāya comparative study of SA 459 gives a good idea of what both suttas exactly mean. (page 31)
The suttas are full of example of the "neither in", nor "apart from"; like in Khemaka sutta (SN 22.89) (or MN1).
Friends, I do not speak of form as ‘I am,’ nor do I speak of ‘I am’ apart from form. I do not speak of feeling as ‘I am’ … nor of perception as ‘I am’ … nor of consciousness as ‘I am,’ nor do I speak of ‘I am’ apart from consciousness.
Rūpaṃ asmīti vadesi, aññatra rūpā asmīti vadesi, vedanaṃ … saññaṃ … saṅkhāre … viññāṇaṃ asmīti vadesi, aññatra viññāṇā asmīti vadesi.
People should learn more about asmi (being). They should equate that with the being "born in this", "present in that" of Aristotle's categories. That is just what Buddha talks about. This is just general, ecumenical philosophy.

“Beings (satta) are owners of their actions, heirs of their actions; they originate from their actions, are bound to their actions, have their actions as their refuge. It is action that distinguishes beings as inferior and superior.”
Sattā kammadāyādā kammayonī kammabandhū kammappaṭisaraṇā. Kammaṃ satte vibhajati yadidaṃhīnappaṇītatāyā”ti. 
MN 135

Some people think that everything is thought (idea); but this is a late Buddhist concept. What is known as Idealism, or the bahya-artha-sunyata of the Gupta era (roughly 4th - 5th century CE)
But in Early Buddhism (pudgala-sunyata), there is body too. And an ultimate reality of the elements. And causality as the functional interdependence of every element (and dhammas) upon all the others; and not as the production of something out of other things (like in Saṃkhya). Elements and dhammas come (to be) together into play (sambhavanti). Yet a self that is unreal, because it comes from nothingness. 

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The Vedic notion of nāmarūpa-vyākaraṇa is ingrained in Buddhism. Otherwise Buddha would have renamed nāmarūpa. We all know the propensity of Indians at renaming what has a slight change in quality. Isn't Brahma becoming Prajapati or Iṣvara for trivial reasons?

The illusive nature of the cosmos is nonetheless real says nāmarūpa-vyākaraṇa. Both Vedism & early Buddhism agreed on that. The difference is how both schools saw maya, the illusion.
Suppose, bhikkhus, that a magician or a magician’s apprentice would display a magical illusion at a crossroads. A man with good sight would inspect it, ponder it, and carefully investigate it, and it would appear to him to be void, hollow, insubstantial. For what substance could there be in a magical illusion?
SN 22.95
The illusion in Buddhism comes from the fact that things come from nothingness, says Buddha. There are “No Things” in saṅkhāra nidāna or viññāṇa nidāna. But there are things (and real things,) in the rūpa of nāmarūpa.
It is not an Upaniṣadic illusion, where things “are not what they really are”; but an illusion that, “according to what have become” (yathabhūta), the things have come from nothingness.
The soteriology is not in the Vedic Upaniṣadic Brahmanic view of “things as they really are”; but in the Buddhistic view of “how (impermanent) things have become” and how “they have become” from nothingness. Bhūta is the past participle of bhavati. Not a present in the now.
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“How (impermanent) things have become” is not a concept that was common before Buddha awakened. (SN 12.65)
Now we do know those facts - and they seem evident to us; but that revelation was far from being around before Buddha's awakening.
The realness of rūpa (of which satta (man/woman) is a part), makes the actualisation complete.
Otherwise, we would have to go to another absurd scenario, with gods made flesh, etc., etc. 
Buddha's view of a real nāmarūpa (like Vedist India's view), settles that unnecessary additional step.

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Are we responsible?

There are hundred of suttas where atta is mentioned
I will just take AN 3.40 (with parallel):
For one performing an evil deed
there is no place in the world called “hidden.”
The self within you knows, O person,
whether it is true or false.
Natthi loke raho nāma,
pāpakammaṃ pakubbato;
Attā te purisa jānāti,
saccaṃ vā yadi vā musā.
This excerpt answers the question of the responsibility of the real sentient being that is satta, and its pseudo/paradoxical-self.
Read again both parallels of the comparative study of SA 459 /AN 6.38(page 31) .
That should prevent some erroneous interpretation about “there are no people” and “we are not responsible of our acts – it is just dhammas coming together”. 
These excerpts are pretty straightforward about the relationship between satta and nāmarūpa.

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The Now

Buddha did teach about considering the now. But if the now is to recall past events; then this is the now.
If the now is the present event; then this is the now.
It is all about seeing the arising - keeping on + altering - and fading of these things that have come to be. 
And that lasts as long as it lasts.

You should know how things have come to be, then you should see their arising, continuance + change, and their fading.

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Additional thoughts