There is absolutely no question about this 5D3 low light AF assist performance issue with either 580EXII or 600EX-RT. It's real. It's embarrassingly slow. In some situations it's approaching professional suicide.
Everything else about the 5D3 AF is stellar and a huge improvement over the 5D2 & 5DC, but this particular issue is nothing short of excruciating for photographers and their increasingly impatient subjects. My hope is for the promised 5D3 firmware update scheduled for sometime next year to deliver a solution.
There are suggested work-arounds including switching off AF-assist and the one outlined by the OP. Very clunky and a serious diversion from the creative process of squeezing off the killer shot of that perfect moment.
To quote Peter Finch's character Howard Beale in the brilliant movie Network, Beale galvanizes the nation, persuading his viewers to shout out of their windows "I'm as mad as hell, and I'm not going to take this anymore!".