Obama didn't have attorneys, private or federal, in most of the eligibility cases. Most were filed in state courts with folks like the local Secretary of State as the defendant.
I doubt that is true ("most").
However each one of these cases involved a risk of a legal result, or a preliminary result on a motion, or a discovery order, or other precatory action that would have affected other cases or facts that will be relevant in an ultimate case that reaches the merits. Thus the guy who is driving the legal effort will have lawyers looking over the shoulder of everyone who is on his side of every case.
That is why you usually would see one of the large firms involved in a case which involves a national effort on the other side. You see this in proxy fights and other kinds of national litigation.
And you can't use low end associates because the guy who is actually involved needs to understand the substance of the main argument and all of the elements of your client's position so that he can see the things coming in peripheral actions that will affect the ongoing effort.
In this case, they benefited from the fact that the challenging parties did not have a coordinated national plan for raising the issues in the several different kinds of settings in which you might have reached the merits. In cases like a proxy fight, the cost is much higher because the aggressor party does have one set of lawyers in charge.