Was the Abhidhamma taught by the Buddha?

I think it depends on whether abhidhamma method rightly infers from the sutta method. If it is right then it is the teaching of the Buddha, whether Buddha himself formulated the expression, who did compile it or didn't are trivial questions in comparison.

The abhidhamma method is supposed to be different in expression but retaining the same inferable meaning as the sutta.

One can of course master the sutta method first and compare the two. This is also as i see it the only way to settle this matter of authenticity.

If one can't tell if abhidhamma is fully agreeable/correct, essentially correct, mostly correct, partly correct or entirely worthless then one probably has not mastered the expression & meaning of the sutta sufficiently to make the call. That isn't shameful imo because It is unreasonable to expect that a person should be able 'judge' from the point of view of meaning & expression all of the dhamma principles lest one is a Buddha. That being said it's imo reasonable to expect an Ariya to be able to tell if Abhidhamma is essentialy agreeable or not because Sutta say that he/she is able to discern what is rightly spoken (sarakaani sutta).

What i think is quite unforunate is people rejecting the abhidhamma method without being qualified to do so (being unable to prove fault yet making reservations). They force themselved into a position where they oppose the norm of the surviving Vibhajavadin traditions and beyond that have no allies of their own. They are aligned with others who reject Abhidhamma but that may be the extent of their alignment, thus they might find themselves on the outside and alone. In theory this might be totally natural & fine if they had grasped what no1 else could but what are the chances of that happening...

As to the lore of it's origin i don't have anything valuable to say.

From this YouTube video entitled "The Theravada Abhidhamma with Bhikkhu Bodhi (Class #1, 5 Mar 2018)", Ven. Bodhi, the famous translator and scholar of the Pali Canon, explained the account of modern scholarship, which he accepts to be more accurate, compared to the traditional account:

That is the traditional view, but modern scholarship takes a different view. And I have to say that I agree with modern scholarship. The Abhidhamma is a body of teachings which emerged through a gradual process of evolution. ...

The word "Abhidhamma" appears in the Sutta Pitaka, the collection of discourses, but we never see in the Sutta Pitaka a statement to the effect of the Buddha teaching the Abhidhamma to the monks. We never find anything like that, but rather the word "Abhidhamma" always occurs in the context of discussions that monks are holding among themselves. To support this point, I collected a few passages.

First, he quotes from Mahāgosiṅga Sutta (MN 32):

“Reverend Sāriputta, it’s when two mendicants engage in discussion about the teaching. They question each other and answer each other’s questions without faltering, and their discussion on the teaching flows on.

“Idhāvuso sāriputta, dve bhikkhū abhidhammakathaṃ kathenti, te aññamaññaṃ pañhaṃ pucchanti, aññamaññassa pañhaṃ puṭṭhā vissajjenti, no ca saṃsādenti, dhammī ca nesaṃ kathā pavattinī hoti.

After that, he quotes other suttas, but did not mention the sutta numbers, or perhaps I missed it.

So, according to Ven. Bodhi, the Abhidhamma was taught not by the Buddha but arose out of the discussions and analyses of the monks. He suggested that Ven. Sariputta had a strong contribution towards it i.e. he probably started it, but it evolved over time.

From pages 70 - 71 of the book "The Dawn of Abhidharma" by Bhikkhu Analayo:

In the Mahāgosiṅga-sutta and its Madhyama-āgama parallel the reference to “abhidharma talk” or to discussing the “abhidharma” occurs alongside “Dharma talk” or “teaching the Dharma”. This gives the impression that the two terms Dharma and abhidharma are here interchangeable. In the Mahāgosiṅga-sutta, the prefix abhi- would thus convey the sense of “about” or “concerning” the Dharma. The passage would then describe having a talk “about the Dharma” and discussing “about the Dharma”, abhidharma.⁴⁹

The notion that the prefix abhi- conveys a sense of superiority appears to reflect a later understanding of the implications of the term. In line with later understanding, the commentary on the Dhammasaṅgaṇī, the Atthasālinī, refers to the present passage in support of the authenticity of the Abhidharma-piṭaka as the Buddha’s word.⁵⁰

A discourse in the Aṅguttara-nikāya describes a group of elder monks seated together “talking abhidharma talk”.⁵¹ In this case the Madhyama-āgama parallel does not employ the term abhidharma at all, but rather describes that these monks were “wanting to settle a matter of dispute, namely to discuss what is Dharma and Vinaya, what is the Buddha’s teaching”.⁵² This case would be in line with the impression that abhidharma talk can simply stand for talk about the Dharma.