Well, you had at times three empires, or rather an empire split in three rather than in two (as they were still considered as part of a same political ensemble).
That said, they weren't randomly split : usually, they formed coherent political-military ensemble, that could be funded thanks to their own fiscal ressources.
And that's where Africa kick in : see, it was one of the most financially safe provinces of Western Romania, the only that remained (up to the 430's) untouched by the raids, piracy, civil wars, etc. that plagued the Late Empire.
Separating Africa from WRE or a "Central"RE would have eventually meant abandoning these ones.
You always have the possibility of an usurper split off, but I doubt it would easy : as said, Africa was too damn important politically and financially to be let wandering about on its own.
And if you add to Africa Egypt and Cyreanica and Syria...Then you not only handicapp WRE but as well ERE : basically the whole northern, politically vital part of the Empire.
this and would they be better prepared to deal with the Islamist Hordes?
Seriously? I mean, for
real?