I disagree with you here on most of these points. Firstly, the Tetrachy- yes, it did work well for the Empire, because despite occasional flareups of violence at the top, it more or less was able to end the crisis of the third century and open the way to a period of marked prosperity in the fourth. The Persian threat was contained and stopped from permanently overrunning Syria and Armenia, the Germanic peoples were kept very firmly north of the Danube, and breakaway regimes such as that of Carausius in Britain could be dealt with much more adequately by the centre.
What you talk about in your second and third paragraphs is exactly what happened with the Exarchates in OTL. True, the Emperor appointed the Exarch, and expected some tax to come out of the Exarchate, but in almost all other regards, the Exarchate was left very much to its own devices. I agree with what you say about a decentralised structure being quite liable to provo
ke civil wars though- look at all the rebellions from the Exarchates in OTL.
To hold off the Arab invasion of North Africa, what the ERE desperately needs is a period of stability, which it simply did not have in OTL. A run of decent Emperors in Constans II, Constantine IV and Justinian II (well, his first reign anyway) never became the advantage it should have been because of these Emperors' tendency to both lose the plot, and to die young. Under the circumstances, it's very surprising that it took the Arabs as long to conquer the
last Roman outposts in North Africa as it did- not until the beginning of the eighth century did the last remnants of Christian control fall in modern Morroco. Contrast this with the fall of Syria and Egypt. I think the Exarchate in seventh century Africa worked as well as it possibly could have done under the circumstances.