Abhidhamma vs Sutta: Did the Buddha teach about the birth of things?

Yes, even all Dhanmas, things:

Manopubbaṅgamā dhammā manoseṭṭhā manomayā Manasā ce paduṭṭhena bhāsati vā karoti vā Tato naṃ dukkhamanveti cakkaṃ'va vahato padaṃ.

Manopubbaṅgamā dhammā manoseṭṭhā manomayā Manasā ce pasannena bhāsati vā karoti vā Tato naṃ sukhamanveti chāyā'va anapāyinī.

See also deedmade body.

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).

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

Perhaps the Abhidhamma uses birth (jāti) as a synonym for arising (uppāda) in the context of things (dhammam).

It's wrong translation.

1. "tesaṁ tesaṁ sattānaṁ" is chatthi-vibhatti (of), not catutthi-vibhatti (for). So, it is "Yā (jāti) tesaṁ tesaṁ sattānaṁ (of the various beings)", it is not "Yā (jāti) tesaṁ tesaṁ sattānaṁ (for the various beings)". It's like "your birth" is not "for birth".

2. The subject of the sentence is "Yā ... jāti sañjāti okkanti abhinibbatti, khandhānaṁ pātubhāvo, āyatanānaṁ paṭilābho". The subject is not satta. It means the Buddha is focusing on describing dhamma (things which assembled then called Satta) in this sentence as same as in AbhidhammaNaya.

3. The Subjective Complement of "Yā" is "tesaṁ tesaṁ sattānaṁ tamhi tamhi sattanikāye". It means Yā-jāti-...-yā-āyatanānaṁ-paṭilābho of (what is called) satta who living surrounded by sattanikāya.

So the whole sentence in SuttantaNaya is the same in AbhidhammaNaya as...

What is the thing of the various beings (sattanam) who living surrounded by sattanikāya is called "bhavapaccayā jāti"? The thing is called "birth thing, being born thing, appearing thing, arising thing, turning up thing, the manifestation of the constituents thing (of mind and bodily form), the acquisition of the sense spheres thing" is what is called "bhavapaccayā jāti".

Tattha katamā bhavapaccayā jāti? 17.2Yā tesaṁ tesaṁ sattānaṁ tamhi tamhi sattanikāye jāti sañjāti okkanti abhinibbatti, khandhānaṁ pātubhāvo, āyatanānaṁ paṭilābho— 17.3ayaṁ vuccati “bhavapaccayā jāti”.

But because "tesaṁ tesaṁ sattānaṁ tamhi tamhi sattanikāye" is not the subject of the question because the question is about the thing. It is the question inside the dependent origination and suffering noble truth for the right view of eight noble path. It must be only thing, not Satta. So, "tesaṁ tesaṁ sattānaṁ tamhi tamhi sattanikāye" is not included in the question part, but it is The Subjective Complement of the answer part ot show "Satto is unreal. It is only thing arising and vanishing".

Are there any Suttas referring to the "jati" of "things" rather than the "jati" of "beings" in relation to dependent origination?

There are uncountable Suttas. For the example:

kenāyaṃ pakato satto kvaci 1- sattassa kārako

kvaci 2- satto samuppanno kvaci 3- satto nirujjhatīti.

...

kiṃ nu sattoti pacceti 4- māra diṭṭhigataṃ nu te

suddhasaṅkhārapuñjo yaṃ nayidha sattupalabbhati

yathā hi aṅgasambhārā hoti saddo ratho iti

evaṃ khandhesu santesu hoti sattoti sammati 5-

By whom has this being been created?

Where is the maker of the being?

Where has the being arisen?

Where does the being cease?

Why now do you assume 'a being'?

Mara, have you grasped a view?

...

This is a heap of sheer constructions:

Here no being is found.

Just as, with an assemblage of parts,

The word 'chariot' is used,

So, when the aggregates are present,

There's the convention 'a being.'

It's only suffering that comes to be,

Suffering that stands and falls away.

Nothing but suffering comes to be,

Nothing but suffering ceases.

Life is easy if we open our view to learn.