My impression was that Arab society was progressing quite nicely until Islam came along. Where am I wrong?
they were (and remain) buried up to their eyebrows in tribalism, blood-feud, folly, and indiscipline.
Under the hellenistic and, later, Romans, some civilizing took place in the levant. The Mede did likewise in their territories. The Ethiopians had some limited effect in the Hadraumat.
However, none penetrated into the interior of the Arabian peninsula, and none really crushed the tribalism endemic to primitive peoples.
The man I blame more than any other for the rise of Islam is Justinian "the Great" - his lunatic reconquest of the western empire bankrupted the East, creating a power vaccum. The plagues which followed (probably due to a volcanic-induced cold snap) didn't help. Additionally, in order to gain the support of the orthodox, alexandrine, and roman churches, he deliberately persecuted the monophysite christians of Syria, the levant, Egypt, and coastal Arabia. This drove these peoples from the church, and primed them for conversion to an indigenous creed which appealed to the austerity they had already developed.
All of this is Justinian's fault.
Had he abandoned the West, come to some form of agreement with Khosroes of Persia, and embraced the real strength of his empire (the "heretic" peasantry of the levant), Rome wcould have endured (flourished) to this day, and there certainly would be no Islam.