Not if you are a paranoid Roman emperor (an obvious tautology): a successful delegate can cost you the throne, and unsuccessful one can usually cost you some troops and money
Because the Romans never employed Generals or Governors.
The first attempt to retake Africa was a very costly failure. If there is a second failure, I doubt very much that there will be a third one.
But my point is really that Justinian will not stop: Africa, Sicily, Italy, Hispania.
The gambler is always going for another throw. Will Gaul be the next one?
OTOH, the former WRE is much poorer than the ERE, and population has sharply decreased. Who is going to pay for the defense of a much longer border? Who is going to man it?
In the 6th century the ERE does not need the WRE, and should concentrate on holding the truly important borders: the Balkan one and the Persian one.
Well, Africa and Italia were the main targets - and prior to the damage caused in the Gothic Wars, both Africa and Italia were very wealthy - and would have paid for themselves - and without the Vandals on the sea, the navy wouldn't have to be expanded much because there was no other naval power to worry about.
So we're taking the choicest provinces. Spain is next on this list, and capturing Spain would make the borders to control smaller - whilst allowing access to Spanish metals that would benefit the Empire. The only territory left is Gaul - and defending the Alps and Pyrenees is a pretty decent border IMO.
Now - regarding the wealth - sure the west isn't as wealthy - but it is still wealthy enough to pay for its own defense - its biggest shortfalls before 453 were tax revenues were practically non-existent in Italia, and manpower.
Now the manpower problems will still have to be resolved through conquered Goths and Vandals - not impossible, but if these new troops are dispersed across all fronts, then they can't form a coherent force to be an issue. - financially, the ERE with its financial reforms, and its money, can back the west, safer trade in the Med because there are no vandals and fewer pirates will also provide revenues to pay for soldiers.
The issue with Justinian isn't that he's a compulsive gambler - he's an opportunist on a mission to reunite the Roman Empire. That combined with the typical behavior of an Empire (Expansion) means that this would happen under any opportunist. In fact, cowering in one half of the Med may end up being bad for the Romans, as the West gets stronger.