It's not that that I've not heard the 'exhaustion' explanation stated before, but that I think it results from a [possibly inappropriate] metaphor rather than analysis. The wars involved increased militarization in precisely the regions that the Arabs conquered, it did not reduce militarization. The problem, as I see it, was that this militarization took its form, in the south, from Arab clients.
Arabia was not, as you said, united. Arab unity involved military success and was, many would presume, held together by booty as much (if not more than) religion, the booty coming from the conquests; the unification of the Arabs was a product not a cause of the conquest. On the other hand, the unification of the Hijaz created a military unit large enough to 'get started', whose main rivals were those Persian and Roman clients and, in defeating them, left much of the south with limited immediate ability to defend itself. It still wouldn't have mattered I don't think if the neighbouring populations had resisted more thoroughly, but their willingness to do so was short due, in Roman land, to being oppressed by Constantinople's attempts to impose imperial Christianity at the expense of local variants; and in Mesopotamia and Khuzistan, from being Christians ruled by ambiguous Zoroastrian overlords.
If 'exhaustion' played a role, it might have been in generating consent elsewhere in the Roman Empire to raise armies big enough to retake the lost provinces?
