Simply put, the 6D outperforms the 5D3 for low light focusing.
Having neither access to a 5d3 or 6d, but wanting to decide for one sooner or later it's rather difficult because it's very fuzzy what "low light" is (few people posting state their lens, object distance + shutter&aperture) - and how fast either camera focuses with af assist disabled in lower light.
On the one hand, I am amazed how many people have this problem w/o an official Canon confirmation, on the other hand if the issue is really as severe as it appears to be for "5d3-like" situations there should be an even larger outcry - or Canon really changed something on the latest hardware?
Summing it up, I understand this is an important issue for lower-light shooting like events and receptions where even the up to -2lv 5d3 needs the af beam, and there is no solution in sight except probably using non-Canon flashes, correct?
It's a frustrating issue for the reasons you state. It's a moving target kind of intermittent problem because it seems like some folks have dealt with it in a big way, others aren't sure (or don't use or care about low light shooting) and still others don't see any problem whatsoever and seem to think the low light AF works better than anything they've ever seen. I had a major problem with it myself, exchanged the camera and I now think it merely works OK, it's not a total show stopper anymore but it doesn't blow me away either.
I would suggest you rent or borrow both the 5D3 and the 6D and see for yourself based on how/what you shoot. For me, I wouldn't have paid much attention to the 6D if the 5D3 had worked as well as I thought it should have in low light. Focus hunting and missing shots doesn't = 'working well' to me. As things stand at this point in time, my use bears out that the 6D works better in low light. YMMV.
As for using non-Canon flashes, setting tweaks, etc I can't give you a good opinion because none of it matters in my use. The functionality was the same in general no matter how I tried to help the problem. I guess the AF Assist from a Yongnou 622C helps a little but I don't use AF Assist so it doesn't matter to me. (I do have the 622C units, they are pretty neat but the beam cluster is a bit off center.) The small SunPak RD2000 flash I have used for years doesn't have AF Assist, won't work with the 622C and I'm not going to start using a big flash just to gain the AF Assist beam that I never needed in the past with my 5Dc. Plus, I don't like distracting people with flashing things in low light. If you read my numerous posts earlier in this thread, I don't want to trade the embarrassment of poor AF delays for the embarrassment of flashing distractions in low light. Just what I want in a church service, flashing patterns on the pastor, alter and front of the sanctuary. (Now that I have a silent shutter!)
Sorry for the length. Feel free to share your thoughts and experience and contact me if you like. Good Luck!