84westYou bring up a lot of points so I will try and address them in some kind of order. First, buying a camera that can do both photo and video at a high commercial level is kind of an oxymoron. That being said, still cameras that can also shoot video are a thing, while video cameras that can shoot video are not. You need to think about which you do more, or more importantly which makes you the majority of your income.
You also need to think about budget, You didn't mention the amount you are willing to spend or any lenses so I can't offer much help there, but i can say that if you are heavily invested in Canon glass then switching systems may not be ideal. You also mention wanting 4K or high frame rate 1080p, but if you don't have nice glass that resolution is wasted. (I also would take 10-bit color and 1080 over 8-bit and 4k on almost any shoot, but that's perhaps a separate conversation.) If you are heavily invested in Canon glass then that probably limits you to Canon and Sony for the most part. The only reason I see to upgrade your body for video is if you have clients where 1080 instead of 4K is an absolute deal breaker or you need a higher frame rate (for shooting skiing for example.)
I've shot a lot of sports and if it is a priority for you I would stick with a DSLR (and add that a 6D really isn't a good autofocus system to hold up as the example of low light performance.) But again, lenses are more important, a 70-200 or 300 2.8 on just about any body will out perform a 24-105 or similar kit lens. I haven't used a sony in low light a ton, but I did find that an a7II with adapted glass performed about as well as a 6D for concert shooting, but significantly worse then a 1DX or 1D mk4.
You have a ton of options and you cheapest cost to image quality gained option will almost always be keep your body and get nice glass. If you shoot mostly video and sometimes stills i'd look at the A7 line and an adapter for any canon glass you have. If you shoot both about equally I would keep the 6D and look for a used C100 (and some nice audio equipment). If you shoot mostly stills or don't have a couple thousand bucks minimum to really buy in to a system I would keep the 6D and invest in glass. GH4/5s are popular on here because they are a compact high res/high frame rate option for a low(ish) cost that makes them excellent for shooting skiing, but I wold be wary of that small of a sensor and the limited native lens options for stills especially indoor sports shooting.
You seem to know exactly what's going on here so I'll address some of what you said
1. In terms of income its a pretty solid 50/50, I'm known more as an overall media guy/content creator to most of my clients.
2. In terms of glass I'm pretty heavy into Canon. 70-200 2.8mkll and a sigma 24 1.4, both of which I love and don't really want to switch from.
3. I completely understand what you're saying about the 1080/4k and bit rate dilemma, I'm sure you understand how clients can be these days though. I've had people hit me up to do a video for Instagram but long story short they didn't end up hiring me because I couldn't do 4k. For Instagram...
4. I know the 6D isn't a good autofocus system but it's been getting the job done for me, just wanting to make sure whatever I switch to is equally good or better. That's also the reason I wanted to stay away from GH4/5.
I hadn't thought about the C100 option yet, I'm gonna look into that now.
Any more input is appreciated