Was the Abhidhamma taught by the Buddha? According to the Theravada tradition, the Abhidhamma was taught by the Buddha.

From "The Abhidhamma in Practice" by N.K.G. Mendis:


  Theravaada tradition holds that the Buddha conceived the Abhidhamma in
  the fourth week after his enlightenment, while still sitting in the
  vicinity of the Bodhi tree. Tradition also has it that he first
  preached the Abhidhamma to the assembly of deities in the Taavatimsa
  heaven; his mother, reborn as a deity, was present in the assembly.
  This can be taken to mean that the Buddha, by intense concentration,
  transcended the earth-bound mentality and rose mentally to the world
  of the deities, a feat made possible by his attainment of higher
  powers (abhiññaa) through utmost perfection in mental concentration.
  Having preached the Abhidhamma to the deities, he returned to earth,
  that is, to normal human consciousness, and preached it to the
  venerable Saariputta, the arahant disciple most advanced in wisdom.


However, in this YouTube video, Ajahn Brahm stated that the Theravada Abhidhamma was not taught by the Buddha and that the complexity of the Abhidhamma is not needed. He said that the Buddha's original teachings in the Pali suttas is simpler and completely sufficient.

One of the evidences he cited is as follows - only the Suttas (Dhamma) and the Vinaya were recited at the First Buddhist Council and there was no mention of the Abhidhamma.

I quote from Harvey, P (2013), "An Introduction to Buddhism: Teachings, History and Practices" (2nd Edition), Cambridge University Press, p. 88:


  Just after the Buddha’s passing away (c. 404 BCE), a 'communal
  recitation' (council)  of 500 Arahats was held at Rājagaha (Skt
  Rājagraha, Vin.ii.284–7) to agree the contents of the Dhamma and
  Vinaya which the Buddha had left as ‘teacher’ (D.ii.154). Ānanda,  the
  Buddha’s  faithful attendant monk, recited the Suttas, such that each
  begins: ‘Thus have I heard’. The monk Upāli recited the Vinaya.


This wikipedia page on Theravada Abhidhamma states that according to scholars, the Theravada Abhidhamma was probably composed in 3rd century BCE, which is after the Buddha's lifetime. I also call it the Theravada Abhidhamma specifically, because there seems to be another Sarvastivada Abhidharma.

So, my question is: Was the Theravada Abhidhamma really taught by the Buddha? Or is that an exaggeration?
Mendis refers to “Theravāda tradition”. His source is the introduction to the Atthasālinī, which was compiled by Buddhaghosa in about 400 AD based on earlier texts that no longer exist.

The Abhidhamma is a framework that consolidates teachings from more than 10,000 Suttas. This framework follows the Theravāda perspective. IMHO, the scholar’s view that the Abhidhamma was developed slightly after the Suttas makes sense.

Though the Abhidhamma may not be the literal word of the Buddha, it is extremely important. The commentaries, which explain the Suttas according to the Theravāda perspective are based on the Abhidhamma.

I once met a monk who had heard that I taught Abhidhamma. The monk said, “I have my own interpretation of the Suttas based on my own experience. It differs from the Abhidhamma and Commentaries.”

I replied, “Venerable Sir, I cannot say whose interpretation is correct. But if I follow your interpretation, then if I have any questions, I have only one source of reference... you. If I follow the traditional Theravāda interpretation, then I can refer to many books and many teachers.”

People who say, “I will only focus on the Suttas and ignore the Abhidhamma and Commentaries” risk misinterpreting the Buddha’s message! Unfortunately, I have seen many examples of this.