How is the "manifestion" (pātubhāvo) of the earth element the arising (uppāda) of suffering?

The four key Pali words in this sutta are uppādo, ṭhiti, abhinibbatti and pātubhāvo. Pātubhāvo is used twice, namely, in respect to the object of the sutta (eg. earth element) and to the outcome, namely, jarāmaraṇassa pātubhāvo (manifestation of aging & death). Therefore, I think to understand SN 26.9, these four Pali words need to be understood; not necessarily according to their dictionary meanings; but according to how they are used in the Pali suttas.

In the Pali suttas, pātubhāvo is used in the following contexts:

Monks, the manifestation (pātubhāvo) of six things is rare in the world, namely: 1. The manifestation of a Tathagata (Buddha). 2. One who can teach the Dhamma & Discipline of a Tathagata. 3. Attainment as a noble (enlightened) disciple. 4. Endowment with unimpaired sense faculties. 5. Being intelligent & astute. 6. Desire for the wholesome Dhamma. AN 6.96

The manifestation (pātubhāvo) is five gems in rare in the world. What five? The manifestation of a Tathagata; a person that teaches the Dhamma proclaimed by a Tathagata; a person who understands the Dhamma when taught; a person who practises that Dhamma; and grateful & thankful person. It is the manifestation of these five gems that is rare in the world. AN 5.144

When the mind is concentration, the Dhamma becomes manifest (pātubhāvo), because of which he is one reckoned as 'one who dwells diligently'. SN 55.40

At Savatthī. Bhikkhus, these eight things, developed and cultivated, if unarisen do not arise apart from the appearance of a Tathagata, an Arahant, a Perfectly Enlightened One. What eight? Right view … right concentration. These eight things…. SN 45.14

If, friends, internally the eye is intact but no external forms come into its range, and there is no corresponding conscious engagement, then there is no manifestation (pātubhāvo) of the corresponding section of consciousness. If internally the eye is intact and external forms come into its range, but there is no corresponding conscious engagement, then there is no manifestation (pātubhāvo) of the corresponding section of consciousness. But when internally the eye is intact and external forms come into its range and there is the corresponding conscious engagement, then there is the manifestation (pātubhāvo) of the corresponding section of consciousness. MN 28

These five faculties of his— the faculty of conviction, the faculty of persistence, the faculty of mindfulness, the faculty of concentration, the faculty of discernment — manifest (pātubhāvanti) weakly. Because of their weakness, he attains only slowly the immediacy that leads to the ending of the effluents. This is called painful practice with slow intuition. AN 4.163

In the Pali suttas, abhinibbatti is used in the following contexts:

Any kamma, bhikkhus, fashioned through greed, born of greed, caused by greed, originated by greed, ripens (vipaccati) wherever individuality (attabhāvo: lit: 'self-becoming') is produced (nibbattati). Wherever that kamma ripens, it is there that result is experienced, either here & now (diṭṭheva dhamme), later (upapajje) or following that (apare pariyāye). A person unknowing: the actions performed by him, born of greed, born of aversion, & born of delusion, whether many or few, are experienced (vedaniyaṃ) right here (idheva): no other ground is found. AN 3.34

Suppose, bhikkhus, an artist or a painter, using dye or lac or turmeric or indigo or crimson, would create the figure of a man or a woman complete in all its features on a well-polished plank or wall or canvas. So too, when the uninstructed worldling produces (abhinibbattento) anything, it is only form that he produces (abhinibbatteti); only feeling that he produces; only perception that he produces; only volitional formations that he produces; only consciousness that he produces. SN 22.100

By recollecting his ancient maternal & paternal family lineage, his individuality (attabhavassa) is produced (abhinibbatti)... MN 96

In the Pali suttas, uppādo is most prominently used in "paticcasamuppada", namely, "dependent co-arising".

An examination of the above three Pali words shows they refer primarily to mental production, mental manifestation & mental arising. "Abhinibbatti" in particular refers to the production of ideas about "self" or "individuality" ("attabhava"). .

It follows the meaning of SN 26.9 is the continued manifestation, production and arising of views of "self" based on the six elements is the mental manifestation of (ideas) of aging & death that result in suffering.

Most importantly, SN 26.9 show the "physical" interpretation of "birth" ("jati") in the Dependent Origination is wrong (since these Pali words abhinibbatti and pātubhāvo are found in the definition of "jati").

How is the “manifestion” (pātubhāvo) of the earth element the arising (uppāda) of suffering?

In another sutta (MN 140) these six "elements" are said to be elements (also translated "aspects" or "properties") of "this person" -- I guess that some, for example the earth element, are also elements of other perceived objects or sense-objects.

Anyway, (according to SN 25.9) these elements are "impermanent, perishing, and changing" -- so they're unlike the "elements" or "atoms" described by 19th century physics or chemistry, i.e. these "elements" should not be understood to mean "elementary and therefore more-or-less eternal".

In summary I think that they're considered (to be elements or aspects of, properties of) a sankhara, and therefore dukkha (as well as being anicca).

I think the (only) asankhata-dhatu ("unconditioned element") is nibbana.

Since the 2nd Noble Truth and Dependent Origination explain suffering arises due to craving, attachment & becoming, how does the "manifestation" of the mere physical elements, per the above translations, manifest as the arising of "suffering"?

I think that "suffering" isn't the only sense/translation of dukkha -- it's also used to mean "unsatisfactory" -- the dictionary definition also suggests "not what one wants"; an antonym of sukha; and warns that:

"There is no word in English covering the same ground as Dukkha does in Pali. Our modern words are too specialised, too limited, and usually too strong".

So people tend to refer to dukkha instead of the English word "suffering", and/or understand that "suffering" is a reference to dukkha and not necessarily used e.g. in the English sense (where it's used in the Christian canon extensively, to refer to crucifixion for example, among other references).

The formula of SN 26 is:

“Mendicants, the arising, continuation, rebirth, and manifestation of X is the arising of suffering, the continuation of diseases, and the manifestation of old age and death.

The cessation of X is the cessation of suffering, the settling of diseases, and the ending of old age and death.”

If you read the whole of SN 26 (Uppada-samyutta), X refers to eye, ear, nose, tongue, body, mind, sights, sounds, smells, tastes, touches, thoughts, different types of consciousness, different types of contacts, different types of feelings, different types of perceptions, different types of intentions, different types of cravings, the six elements and the five aggregates.

This is just restating "sabbe sankhara dukkha" (Dhammapada 278) or "all conditioned things are suffering" in a longer and more elaborate way, where X = sankhara.

All these conditioned things arise (uppāda), continue (ṭhiti) and reborn (abhinibbatti). These are explained by dependent origination. A "person" or "individual" is compounded and composed of the aforementioned conditioned things. Dependent origination explains the details of how that works.

The fact that they arise (uppāda), continue (ṭhiti) and are reborn (abhinibbatti), would be summarized as their manifestation (pātubhāvo).

Arise here is when the conditioned things arise at birth and continue throughout the moments of life. Rebirth here indeed refers to death and break-up of the body, followed by rearising or rebirth into a new life. Altogether, these stages would represent the manifestation of these conditioned things. The very manifestation of these conditioned things are suffering. This is how it is related to "sabbe sankhara dukkha".

How can I support my opinion that the word "abhinibbatti" indeed refers to "literal" rebirth here? Well, we just need to see how it is used in the other suttas:

In MN 96:

But they are reckoned by recollecting the traditional family lineage of their mother and father wherever they are incarnated.
Porāṇaṃ kho panassa mātāpettikaṃ kulavaṃsaṃ anussarato yattha yattheva attabhāvassa abhinibbatti hoti tena teneva saṅkhyaṃ gacchati.

If they incarnate in a family of aristocrats they are reckoned as an aristocrat.
Khattiyakule ce attabhāvassa abhinibbatti hoti ‘khattiyo’tveva saṅkhyaṃ gacchati;

And as Piya Tan explains for MN 96 here:

Porāṇaṁ kho pan’assa mātā,pettikaṁ kula,vaṁsaṁ anussarato yattha yatth’eva attabhāvassa abhinibbatti hoti tena ten’eva saṅkhyaṁ gacchati. “(His) physical rebirth,” attabhāvassa abhinibbatti, lit “production of selfhood.” Atta,bhāva (BHS ātma,bhāva), bodily form, body; existence as an individual; living being (V 2:238,17 = A 4:200,6 = 204,2 = 207,2; D 3:111,10; M 2:32,8, 181,11; S 5:442,1; A 1:279,2, 3:411,23; J 4:461,25; Ap 215,11; Miln 171,- 13; Vism 310,27. Often as ~paṭilābha, the becoming; reborn as an individual; reincarnation; type of body or exist- ence (V 2:185,25 = A 3:122,24; V 3:105,20 = 107,35 = S 2:255,19; D 3:231,16 = A 2:150,5; M 3:46,6 = 52,33; S 2:272,4, 283,33, 3:144,12); A 2:288,30. While puna-b,bhava (D 2:15; S 1:133, 4:201; Sn 162, 273, 502, 514, 733; It 62) is the term for rebirth as a cycle, abhinibbatti refers to a particular rebirth, usu in a physical form (ie a reincarnation) (D 2:305,7 = M 3:249,16 = S 2:3,7 = Vbh 99,14; D 3:94,28): see CPD sv. Also common is punabbhavâbhi- nibbatti, “rebirth in a new existence” (M 1:294; S 2:65; A 1:225; V 3:3).

Bhikkhu Sujato states here that:

abhinibbatti = rebirth (pretty much only used in this sense)

According to OP:

Bhikkhu Bodhi footnote: 'attabhāvassa abhinibbatti' literally should be: "wherever the reconception of his individuality takes place"

In the book "Investigating the Dhamma: A Collection of Papers" on page 52, Bhikkhu Bodhi elaborates on this, explaining that "conception" here literally means conception into a new life or new birth:

In fact, the word 'abhinibbatti' is used as one of the synonyms of jāti in the standard definition of the latter. Apparently when abhinibbatti is included in jāti we should understand jāti as comprising both conception and physical birth, while they are differentiated, abhinibbatti means conception and jāti is restricted to full emergence from the womb.

The genius student can memorize and understand a big/maximum data each studying, so, the sutta is long. But, if the student is not too much smart, he can memorize and understand a little/minimum data each studying, so the teacher have to teach the sutta for him in short form and maybe incomplete paṭiccasamuppāda's process.

Because, in the oral study system, we memorize one sentence/paragraph (=each sutta) from the buddha or the tipitaka-memorizer (arahanta). Then we study/ask/practice/comprehension follow that memory, until we perfectly understand it (professional). If we still not enlighten after that. We will go to memorize the other lesson more, again. We will loop it until we are the tipitaka-memorizer, too. This is the tipitaka tradition. We still doing follow this tradition, for 2600 years.

So, it is already described itself, if you read the previous sutta. They were sequenced by paṭiccasamuppāda.

Let's see:

13.4.1 (30) Catusso — Four Elements = kevala(ssa) dukkhakkhandha(ssa) = rūpa-paṭiccasamuppāda, upattibhava/jāti/jarā/maraṇa-paṭiccasamuppāda.

13.4.2 (31)/13. 4. 3.(32) Pubbe — the satisfaction (assāda), danger(ādīnava) and the escape (nissaraṇa) on account of the earth element.

Assāda = 60 piyarūpa-sātarūpa, in saccapabba of Sutta. Dī. Ma. mahāsatipaṭṭhānasutta, which are objects of taṇhā-paṭiccasamuppāda:

Bodily phenomena in the world are pleasant and agreeable, there taṇhā, when arising, arises, there when settling, it settles.

Buddha said assāda as somanassa-vedanā-paṭiccasamuppāda in these 2nd and 3rd sutta, 13. 4. 2. (31)/13. 4. 3.(32):

Whatever pleasantness and pleasure arises on account of the earth element, that is the satisfaction in the earth element.

So, above quote showed assāda as 60 piyarūpa-sātarūpa which come to salāyatana-paṭiccasamuppāda, then let somanassa-vedanā-paṭiccasamuppāda arise, then let taṇhā-paṭiccasamuppāda arise.

But the way to escape "kevala(ssa) dukkhakkhandha(ssa) in paṭiccasamuppāda" showed in these 2nd and 3rd sutta, 13. 4. 2. (31)/13. 4. 3.(32), too:

“ That the earth element is impermanent, unpleasant and a changing thing is the danger (ādīnava) in the earth element. Taming the interest and greed for the earth element and dispelling the interest and greed for the earth element is the escape(nissaraṇa) from the earth element.

13.4.4 (33) Yo no cedaṃ –– If There Weren’t, is then if the practitioner can not realize the danger (ādīnava) in the earth element to escape (nissaraṇa) it, then there are the interest and greed (taṇhā/upādāna-paṭiccasamuppāda) which attaching somanassa-vedanā-paṭiccasamuppāda. (And that sukkha/somanassa-samphassaja-vedanā-paṭiccasamuppāda taking the earth element at kāya/mano-āyatana-paṭiccasamuppāda together with kāya/mano-samphassa-paṭiccasamuppāda as well. Also, that kāya/mano-āyatana-paṭiccasamuppāda are rūpa/nāma-paṭiccasamuppāda, which is "mano pubbaṅgamādhammā".) So, buddha showed somanassa-vedanā-paṭiccasamuppāda, taṇhā/upādāna-paṭiccasamuppāda, and upattibhava-paṭiccasamuppāda in 13. 4. 5. (34), 13. 4. 6. (35):

6. “Bhikkhus, he who does not delight (taṇhā-paṭiccasamuppāda which attaching somanassa-vedanā-paṭiccasamuppāda) in the earth element (mana-āyatana-paṭiccasamuppāda's object) does not delight in unpleasantness(kevala[ssa] dukkhakkhandha[ssa]). He that does not delight in unpleasantness, I say, is released from unpleasantness.

Then the buddha presented the effects, jāti/jarā/maraṇa-paṭiccasamuppāda, of "he who does not delight in the earth element does not delight in unpleasantness. " from 13. 4. 5. (34), 13. 4. 6. (35) into 13. 4. 7. (36) Uppado — Arising:

2. “Bhikkhus, the appearance(upādo=jāti-paṭiccasamuppāda), being (ṭhiti=jāti-paṭiccasamuppāda), arising (abhinibbatti=jāti-paṭiccasamuppāda), and rebirth (pātubhāva=jāti-paṭiccasamuppāda) of the earth element is the appearance of unpleasantness, the continuance of ailments and the manifestation of decay and death.

Please notice, abhinibbatti and pātubhāva are used as the synonym of jāti-dukkha-ariyasacca to describe "jāti" in saccapabba of Sutta. Dī. Ma. Mahāsatipaṭṭhānasutta, too. So, they are connect together.