[gtyruzzel]

in the Buddha's teaching, there is also birth of 'self' due to ignorance

There is the case where an uninstructed, run-of-the-mill person assumes form, feeling, perception, fabrication &/or sense consciousness to be '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 is born of that.

SN 22.81 and there are the arahants, which have ended 'birth'

Now at that time a large number of monks had declared final gnosis in the Blessed One's presence: "We discern that 'Birth is ended, the holy life fulfilled, the task done. There is nothing further for the sake of this world.'

SN 12.70 When the currents of construing do not flow, he is said to be a sage at peace. 'I am' is a construing. 'I am this' is a construing. 'I shall be' is a construing. 'I shall not be' is a construing. By going beyond all construing, he is said to be a sage at peace.

Furthermore, a sage at peace is not born, does not age, does not die, is unagitated and is free from longing. He has nothing whereby he would be born.

MN 140 ***