Does it make sense to talk about origin of self?

In dependent origination, the stage self comes into picture is at 'attachment' and is full mature at 'birth'. As a modern Indian, you (the questioner) should know the word 'jati' ('birth') does not mean 'physical birth from the mother' but means 'self or social identity'.

Jāti (in Devanagari: जाति, Bengali: জাতি, Telugu:జాతి, Kannada:ಜಾತಿ, Malayalam: ജാതി, Tamil:ஜாதி, literally "birth") is a group of clans, tribes, communities and sub-communities, and religions in India. Each jāti typically has an association with a traditional job function or tribe. Religious beliefs (e.g. Sri Vaishnavism or Veera Shaivism) or linguistic groupings may define some jatis. A person's surname typically reflects a community (jati) association: thus Gandhi = perfume seller, Dhobi = washerman, Srivastava = military scribe, Ror = Warriors etc. Wikipedia

The self certainly originates, as described in many suttas below:

And what is clinging? These four are clingings: sensuality clinging, view clinging, precept & practice clinging, and doctrine of self clinging. This is called clinging.

SN 12.2

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The craving that makes for further becoming — accompanied by passion & delight, relishing now here & now there — i.e., craving for sensual pleasure, craving for becoming, craving for non-becoming: This, friend Visakha, is the origination of self-identification described by the Blessed One.

MN 44

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There is the case where an uninstructed, run-of-the-mill person — who has no regard for noble ones, is not well-versed or disciplined in their Dhamma; who has no regard for men of integrity, is not well-versed or disciplined in their Dhamma — assumes form to be the self. That assumption is a fabrication. Now what is the cause, what is the origination, what is the birth, what is the coming-into-existence of that fabrication? To an uninstructed, run-of-the-mill person, touched by that which is felt born of contact with ignorance, craving arises. That fabrication [of self] is born of that. And that fabrication is inconstant, fabricated, dependently co-arisen. That craving... That feeling... That contact... That ignorance is inconstant, fabricated, dependently co-arisen.

SN 22.81

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This, monks, is the path of practice leading to self-identification. One assumes about the eye that 'This is me, this is my self, this is what I am.' One assumes about forms... One assumes about consciousness at the eye... One assumes about contact at the eye... One assumes about feeling... One assumes about craving that 'This is me, this is my self, this is what I am.'

"One assumes about the ear...

"One assumes about the nose...

"One assumes about the tongue...

"One assumes about the body...

"One assumes about the intellect that 'This is me, this is my self, this is what I am.' One assumes about ideas... One assumes about consciousness at the intellect... One assumes about contact at the intellect... One assumes about feeling... One assumes about craving that 'This is me, this is my self, this is what I am.'

MN 148

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Who, O Lord, has a sense-impression?"

"The question is not correct," said the Exalted One.

"I do not say that 'he has a sense-impression.' Had I said so, then the question 'Who has a sense-impression?' would be appropriate. But since I did not speak thus, the correct way to ask the question will be 'What is the condition of sense-impression?' And to that the correct reply is: 'The sixfold sense-base is a condition of sense-impression, and sense-impression is the condition of feeling.'"

"Who, O Lord, feels?"

"The question is not correct," said the Exalted One. "I do not say that 'he feels.' Had I said so, then the question 'Who feels?' would be appropriate. But since I did not speak thus, the correct way to ask the question will be 'What is the condition of feeling?' And to that the correct reply is: 'sense-impression is the condition of feeling; and feeling is the condition of craving.'"

"Who, O Lord, craves?"

"The question is not correct," said the Exalted One. "I do not say that 'he craves.' Had I said so, then the question 'Who craves?' would be appropriate. But since I did not speak thus, the correct way to ask the question will be 'What is the condition of craving?' And to that the correct reply is: 'Feeling is the condition of craving, and craving is the condition of clinging.'"

"Who, O Lord, clings?"

"The question is not correct," said the Exalted One, "I do not say that 'he clings.' Had I said so, then the question 'Who clings?' would be appropriate. But since I did not speak thus, the correct way to ask the question will be 'What is the condition of clinging?' And to that the correct reply is: 'Craving is the condition of clinging; and clinging is the condition of the process of becoming.' Such is the origin of this entire mass of suffering.

SN 12.2