This issue is a strange one..... I have tested a 5DIII (paired with a 24-70 MII) with the 600rt in a very dark indoor environment and found it to be equally fast (or slow) in acquiring and confirming focus on either single or multiple focus points with and without the aid of the assist beam.
I have one of the first produced 5DMIII's with the light leak (6th digit is 1)
Do you have any other lenses you could test? My 70-200 f/2.8L IS II is the one that gives me issues. My 24mm 1.4L does just about the same with or without the AF assist beam. With the 70-200 there is a very noticeable difference.