Universal Circuits in Indistinguishability Obfuscation Candidate Construction
As @D.W. guessed, the branching program for a circuit essentially reveals the original circuit.  It's not clear what you mean by "apply the whole obfuscation process to the circuit-revealing branching program," but the prospects for that do not seem good: evaluating the branching program is highly sequential (polynomial depth), and you would need to obfuscate that procedure.  In all, you've reduced the goal of "obfuscate a log-depth circuit" to "obfuscate a poly-depth circuit" – that's negative progress!

By contrast, the universal circuit has a fixed branching program, so it obviously reveals nothing about the circuit to be obfuscated (except an upper bound on its size).  Also, the UC's input (the circuit to be obfuscated) is "concealed" by the multilinear map.