I have both the 6D and the 7DII. While there is certainly a big difference between the two as far as auto focus capability, the 6D does a superb job when used in the manner for which it was designed. Landscapes, portraits, macro, anything that doesn't move too fast, etc. I have shot BIF with it with good results, but it is far more challenging then with the 7DII. The speed and accuracy of the auto focus is excellent, but it is just not designed for high speed subjects. On one occasion I used it, with Al Servo and single point focus at one of my nephew's soccer games and it did quite well, but I normally use the 7DII for that with better results. For low light it blows away the 7DII. I often use the 6D, EF 16-35 f/4 L, and no flash for indoor shots with very good results.
It's like anything else, use it in the manner for which it was designed and it is unlikely that you will be disappointed. If it fits your shooting style, then go for it, IMO.

I've never used the Sigma or an extender so I can't help you there, but I do have both the 100-400 L II and the 70-200 f/2.8 L II. I use them on my 6D and 7DII. The 100-400 L II is outstanding. Extremely quiet and quick to focus. Nearly instantaneous actually. Image quality is on a par with the 70-200. From everything I have read, the 1.4x does little if anything to degrade performance. Plus, you could use it on the 70-200 if desired. The 100-400 focuses very close. Closer then the 70-200. I find that I use it more then the 70-200. I generally keep the 70-200 on my 6D and the 100-400 on my 7DII. Makes for a great combo.

No one mentioned that the AF goes south on the 6D with the extender.

However the 100-400 works fine on the 6D without the extender. Of course on the 7DII AF works with the extender. In fact I bought the 1.4X just for that lens on my 7DII. Well the mkI actually, maybe one day I will trade my Mk I and Tamron 150-600 in and get the 100-400 mkII. Or not, I often wish I had two 100-400s so my 6D and 7DII could have it at the same time.
My Tamron 150-600 does just about as well as the 100-400 when the light is really good and of course it has the 200 mm extra reach. But when the light is iffy I find the 100-400 images better, even when cropped to equal the 600s reach.
I have read that the Sigma Sport is a bit better than the Tamron or Sigma Contemporary but is it enough better to justify the weight and size difference?
I can hand hold the Tamron 150-600 but usually use it on a tripod with gimbal head. The 100-400 I hand hold with a pistol grip screwed onto the tripod mount. I have a strong feeling that the Sigma Sport would be a tripod only deal in most cases for me and most others who aren't young & fit or working out to keep upper body and arm strength up.

I've never used the Sigma or an extender so I can't help you there, but I do have both the 100-400 L II and the 70-200 f/2.8 L II. I use them on my 6D and 7DII. The 100-400 L II is outstanding. Extremely quiet and quick to focus. Nearly instantaneous actually. Image quality is on a par with the 70-200. From everything I have read, the 1.4x does little if anything to degrade performance. Plus, you could use it on the 70-200 if desired. The 100-400 focuses very close. Closer then the 70-200. I find that I use it more then the 70-200. I generally keep the 70-200 on my 6D and the 100-400 on my 7DII. Makes for a great combo.