A complex post with a complex answer. My apologies as this won't be short
Let's take the easier subject first: The camera. Truth be told, as a general rule, the more complex your AF requirements are, the more the "better" cameras will serve you in terms of AF performance, which includes of course consistency. This statement does not mean in any way that a "lessor" body is garbage; more that as your own AF needs evolve, and tracking babies might be one of these situations, the better the odds are that a more advanced body with better AF would serve you well. In my experience, having shot or owned nearly ever Nikon DSLR since the D100 so many moons ago, and having owned the D610 up until a couple years ago, the 39pt AF system in the D6xx range is not one I hold in high regard with tracking ability or AF consistency, which is why I moved on from it. However, it isn't junk either. Unfortunately, moving up to a D850 is going to be an expensive proposition, so you might want to seriously consider renting one of those before making any other further judgements.
Now the complicated stuff: The lens.
A lot of people in this forum like to talk about lenses they'e never shot, but I'll get right to the point and tell you I've not shot the 85/1.4 Sigma EX in far too long, nor long enough, to truly know it's character or strengths/weaknesses these days. However, at 85mm, other than not being able to talk about that one, I have a lot of 85mm experience, as for a short time I owned three (!) 85mm lenses at the same time: The Sigma 85/1.4 Art, the Tamron 85/1.8 VC, and the Zeiss Milvus 85/1.4. I currently only own the Sigma art and the Zeiss. Prior to those, I owned the 85/1.8G, the 85/1.8 AFD, the 85/1.4 AFD, and I've shot with the 85/1.4G as well as the Zeiss 85/1.4 Otus.
From a resolution point of view, which means you'd see more of a difference once/if you moved to a higher resolution body, the sharpest 85mm autofocus lens in Nikon mount at this time is the Sigma 85/1.4 Art, hands down, no question about it. It's notably sharper than any other 85mm AF option. Only the two Zeiss manual focus lenses play in the same court - the 85 Zeiss Otus is sharper by a small bit, and the 85 Zeiss Milvus, stopped down to F/5.6 or so, might be a bit sharper as well. But if resolution is your thing, the 85/1.4 Sigma Art is your lens. Focus wise, it's also far better than most earlier Sigma lenses in terms of AF speed and consistency - it's rare for me to miss a shot with it, at least on my D800E bodies. Bokeh wise, it's going to be in the "good but not world class" territory. The best "bokeh" 85mm lenses I've used are, at #1: The Zeiss 85/1.4 Milvus, #2: The Nikon 85/1.4 AFD, and close behind that, the Tamron 85/1.8 VC and Nikon 85/1.4G, with the Tamron having far less CA. I don't consider the 85mm F/1.8 options to have great bokeh, but at the same time, I wouldn't say they have "horrid" bokeh either - it's more to the negative side than the positive, and from what I've heard about the Sigma 85 EX you used to have, that lens had fairly decent bokeh. So it doesn't surprise me you're not happy with the 85/1.8G bokeh.
The 85/1.8G is a pedestrian, pretty good-but-not-great 85mm lens these days. A couple of years ago, it was the best "value for the money'" lens, but IMO, after testing it side by side for nearly a month with the Tamron 85/1.8 VC, the Tamron is easily and clearly the better choice anywhere near this price range, and actually is the lens I'd suggest you try next. I'd also strongly suggest you rent before buying, as rendering gets personal. The Tamron 85/1.8 VC is the current "best bang for the buck" 85mm lens going - it's almost as sharp as the "best", has better bokeh than the Nikon, focuses a bit faster, and has slightly warmer native color cast. If I couldn't own what I do today, I'd have the Tamron, and I'm incredibly picky.
... which leads us to the tricky part of the discussion. Color. Everyone wants to be an "expert" at evaluating color, and "better color" or "worse color" gets thrown out almost immediately when someone either likes or doesn't like a lens. Yet, the sad truth, that you may not like hearing either, is that the odds are you, and most folks in the forum community, don't own a tool good enough to properly judge/evaluate accurate color visually. Truth be told, in my mind, it's a two thing game here - if you truly want to accurately judge and evaluate color, you own either a hardware calibrated Eizo coloredge monitor, or a hardware calibrated NEC spectraview monitor. Everyone else doesn't get to play. Those are expensive monitors. So most folks don't own one (disclaimer: I own an NEC, a friend owns an Eizo, both of used to have less expensive monitors yet neither of us would ever go back. We'd sell a lens if we had to, just to keep our monitors). So the next best thing is to calibrate your monitor, which at least gets you into a reasonable ballpark of being able to judge color. Better yet is to actually *measure* a reference color using the color RGB display in photoshop, which doesn't lie.
... So, that means most folks are judging/evaluating *relative* color differences within their own world of their monitor (which is funny because so many think that this means they can pronounce some lenses as having "deeper richer color" when they don't own a tool that can even remotely allow such an absolute statement to be stated when we think about it). The next thing we have to realize is that many lenses have different color casts - and the cast *itself* can cause one to, shall we say, make an inaccurate assessment of the color. What we should be doing instead is shooting lenses we wish to compare with the white balance preset to each lens, thus normalizing out, as best as we can, the color cast. But back to color casts. Old Sigma glass had a reputation of being slightly warmer, trending to yellow. Some Nikon glass trends very slightly cool. So now let's think what happens when you take a warmer rendering lens (warm native color cast) and shoot human skin. Should look pretty good. In general, humans do not look great if they are cold in color balance, although for some reason many photographers trend towards cold rendering lenses for humans. In contrast, when you shoot a slightly cooler rendering lens on skin, it doesn't look so good, or perhaps it just looks different than what you were used to if you used to shoot a warmer rendering lens. The point is, the direction of the color cast is either going to a) make things look great if the color of the subject works with the direction of the cast, or b), the opposite. This is why you seriously have to watch out for those who claim, say, a Nikon 35/2 AFD has "warmer, richer colors" than it's competitors, because if that lens is WB normalized with the competitors, there won't be any "warmer/richer" at all, and in fact, the 35/2D is generally not as warm (by measurement) as many of the competitors in terms of native color cast. In short, be careful when someone starts using the color "argument" to defend their lens. Don't be one of those guys - learn how to do a preset WB and get the cast normalized out, and you might find the color of the 85/1.8G is good enough for what you do.
So, to summarize:
1) Consider renting a D850 to see if it's AF improvements help you out
2) Consider renting the Tamron 85/1.8 VC, as it's sharper, has better bokeh, than the Nikon 85/1.8G, and it has a slightly warmer native color cast, which even though as explained above you should learn how to normalize out, might be a hair closer to what you were used to.
3) The Sigma 85/1.4 Art is a stupendously sharp lens with "good/not great" bokeh, but with one of the strongest "cold" native color casts around - again, I preset WB my camera just to this lens when I use it. I tend to think you'd hate it's "default" native color cast for people. It's also extremely heavy and big and I'm not sure it's "you" relative to the Tamron noted above.
4) Learn to WB preset per lens before thinking about lens differences in terms of color. You might find you don't mind the 85/1.8G after all if you do that.
Good luck.
-m
