Does DN2 and MN36 provide an irrefutable statement on the fact of rebirth?

It's a stock answer that appears here and there, seemingly in its current form to help pass the old oral tradition. Rebirth, in that it deals with the mostly 'not here and not now', has little to do with ending stress/suffering in the 'here and now'. It's not that it is inaccurate, nor that it is Essentialist. AN 4.77 https://suttacentral.net/en/an4.77, in stating that the full results of Kamma are unconjecturable (undecipherable?), imples that rebirth is the same, as the two are intrinsically linked. I've always had issues with remembering past lives because again, they are not me, they are not mine. (within the Pali canon too, given MN27, proof requires direct experience, not scriptural evidence)

Buddhism has the Triple Refuge, for which the Dhamma as refuge was declared by the Lord Buddha as follows:

Svākkhāto bhagavatā dhammo sandiṭṭhiko akāliko ehipassiko opaneyyiko paccattaṃ veditabbo viññūhītiviññū: wise.

The Dhamma is well expounded by the Blessed One, directly visible, immediate, inviting one to come and see, applicable, to be personally experienced by the wise.

Since the Dhamma is said to be visible here-&-now, a Buddhist of proper faith takes this as this standard & inspects the teachings according to this standard. To choose blind faith over this standard is to destroy enlightenment & the purpose of Buddhism. For example, about Dependent Origination, MN 38 states:

Do you speak only of what you have known, seen, and understood for yourselves? — “Yes, venerable sir.” - “Good, bhikkhus. So you have been guided by me with this Dhamma, which is visible here and now, immediately effective, inviting inspection, onward leading, to be experienced by the wise for themselves. For it was with reference to this that it has been said: ‘Bhikkhus, this Dhamma is visible here and now, immediately effective, inviting inspection, onward leading, to be experienced by the wise for themselves.’ MN 38

With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, he directs and inclines it to knowledge of the recollection of past lives (lit: previous homes).

The above translation is admitted by the translator to be false, because they include "lit: previous homes" in the translation.

The clear & straightforward meaning of this teaching is found in SN 22.79 Khajjaniya Sutta & Haliddakani Sutta. It does not mean reincarnation. The question is answered & finished.

Imo, translators, such as Bhikkhu Bodhi & Thanissaro, who translates "pubbe nivasa" as "past lives" slander the Tathagata because they each have, on certain occasions, translated "pubbe nivasa" correctly in certain places as "abodes" & "homes"; therefore they have demonstrated knowing the exact meaning of "pubbe nivasa".

He recollects... one birth, two births, three births, four, five, ten, twenty, thirty, forty, fifty, one hundred, one thousand, one hundred thousand,

The word "jati" ("birth") is clearly defined in the suttas (eg. SN 12.2) to be the production of "beings" ("satta"). The word "satta" ("a being") is defined in SN 23.3 and SN 5.10 to be "clinging" and "a view".

"Birth" does not mean past & future lives. When the Buddha recollected his previous births, as explained in SN 22.79, the Buddha recollected when the mind, in the past, ignorantly clung to one or more of the aggregates as "self".

Just as if a man were to go from his home village to another village, and then from that village to yet another village, and then from that village back to his home village. The thought would occur to him, 'I went from my home village to that village over there. There I stood in such a way, sat in such a way, talked in such a way, and remained silent in such a way. From that village I went to that village over there, and there I stood in such a way, sat in such a way, talked in such a way, and remained silent in such a way. From that village I came back home.' In the same way — with his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability — the monk directs and inclines it to knowledge of the recollection of past abodes. He recollects his manifold past abodes ... in their modes and details.

The analogy above shows the teaching is not about reincarnation because the man in the analogy "returns home". Thus, the analogy shows there is a "home village", such as the primal ignorant mind, from which the mind creates ideas about self via the process of becoming & unbecoming.

AN 4.121 Attanu­vada ­sutta explains how one may refrain from an impulse to perform a harmful action by considering the associated danger:

Bhikkhus, there are these four [motivating] fears (/threats). What four?

Fear of blame from the self, fear of blame from others, fear of punishment, and fear of bad rebirth (duggatibhayam, lit. threat of misery).

1. Bhikkhus, what is [motivating] fear of blame from the self?

Here, bhikkhus, someone reflects thus - If I were to misconduct by body, words or mind, won't I blame myself for those actions? He, frightened of blame from the self, dispels [the impulse for] misconduct and develops right conduct.

2. Bhikkhus, what is [motivating] fear of blame from others?

Here, bhikkhus, someone reflects thus - If I were to misconduct by body, words or mind, won't others censor me for those actions? He, frigthtened of blame from others, dispels [the impulse for] misconduct and develops right conduct.

3. Bhikkhus, what is [motivating] fear of punishment?

Here, bhikkhus, someone sees the king punishing a highwayman or a robber in various ways such as whipping, beating with the, jungle rope, poisoned stick. Cutting hands, cutting feet, cutting hands and feet. Cutting the ears, cutting the nose, cutting ears and nose. Putting in the gruel pot, giving the shell tonsure, putting in Rahu's mouth, garlanding with a flaming garland, burning the hands with a torch, beating until the body is like straw, making to behave like an antelope, hooking the flesh, cutting squares out of flesh, burning in an alkaline solution, driving a spike from ear to ear to turn, making like a straw foot stool, sprinkling boiling oil on the body, giving to the dogs to be eaten, raising on a pole until death and also cutting the neck. Then it occurs to him the king punishes the highwayman or the robber in various ways for his evil actions and if I do evil actions the same will be done to me, such as whipping ... and also cutting the neck. So he does not rob, fearing punishment from the king. This is called fear of punishment.

4. Bhikkhus, what is [motivating] fear of bad rebirth (misery)?

Here, bhikkhus, someone reflects thus: Bodily misbehaviour leads to bad hereafter, verbal misbehaviour leads to bad hereafter, and mental misbehaviour leads to bad hereafter. If I misbehave by body, words or mind, I too after death will go to misery, to hell, to a bad state and thinking thus he dispels [the impulse for] misconduct and develops right conduct.

The function of rebirth is to serve as a motivating factor. The name of the sutta is Attanu­vada, "the doctrine of self".

If we interpret any of Gotama's words straightly, then we miss (we leave out) any context, any background.

For pure understanding we need to:

1. understand the level and paradigm of asker, auditory culture, background model (brahmanic, upanishadic, samanic, dravidic... et cetera).

2. have understanding about ancient simple Language - they has no words and sciences, has no language for modeling modern texts with modern meanings. What is the meaning of "my", "his life" not in that context but in Dhamma language?

If he says "Me", does he talk about Soul? :) Rupa dhammas? No. Vedana? Sanna? They are gone with death.

Vinnana is clear, and has no memory without sanna and vedana.

He told about new-generated nama-rupa (puggala, manussa), whose non-mindful vinnana was craving to some kamma by moha → contact phenomena → sankhara; and stupid person are mixing past Vipaka with his own "oh, it is my kamma, my fate, my destiny" ... and catch and prolong vipaka as "kamma in his life" by behavior. So, by this way, any catching kamma from past come to real life of present person by the mind about "My". And person are produce new kamma to the world, into the future.

So, by dhamma, abhidhamma: any kamma - anatta, an-atta. Only non-emancipated people are catching vipaka of kamma. So, Arahants cut the craving by wisdom, cut the belts of stupidness and didn't catch any kamma-vipaka anymore. This is the way to Arahantship.

You bunch affection in the head, you cut affection in the head. All in the mind, not in "external reality in time". Mano-settha, mano-maya. Stupid are bunching "me+some kamma", student are practicing non-bound empathy to any kamma-makers in the world (without any "me" or "not me"), wiseman see internal sankharas (kamma and this kamma-makers in past), arahant cut it. Every step in the head.

But in your citation, buddha teach about one kind of practice. In phrase it looks similar to Anapanasati sutta, Satipatthana sutta and many others, where he gave us methodology teachings: "he directs and inclines..." "He recollects..." - very similar to Metta and Brahmavihara style.

The so called "Secular Buddhists" have their own religion consisting of their own fixed beliefs. One of those fixed beliefs is the belief that there is no life after death. Their typical game is to hijack the Buddhist Suttas to give their own interpretations to somehow map the wording to their fixed secular beliefs. If any Sutta proves too difficult to achieve that purpose, they call them later additions or not the word of the Buddha. So in reality they actually take refuge in their materialist/nihilistic views, not in the Buddha, Dhamma and Sangha.