What theory of colonization best explains why the British Empire took formal control of East Africa?
I would argue that the best explanation for this event must rely on more than one theory of colonization. Aspects of all three theories can be seen in the decision to first rule East Africa as a protectorate and then to make it an official colony.
The two theories that do the most to explain this are metropolitan theories and systemic theories. Metropolitan theories focus on factors having to do with the imperial country. In the case of East Africa, we can see that the decision to take formal control was caused in large part by the failure of the East Africa Company. As this company began to fail, the control of the area was transferred to the government and eventually became more formalized. In this way, factors having to do with England and not East Africa caused the change.
However, we must also look at the fact that there was competition between imperial powers. Systemic, international relations-oriented theories argue that this is very important. Because Germany had possessions that bordered on British East Africa, it became more important for Britain to take direct control of East Africa as competition for colonies heated up.
Thus, the British decision to take formal control of East Africa came about because of factors that are best explained by metropolitan and systemic theories of colonization.