SO... IMHO... rambling.... feel free to ignore...
Yes and No...
The D850 is a very good setup. So think very carefully. You only save weight in the body really. You will likely still carry the same bag or hang it on the same strap.
What I am about to say is based on my experience moving from Canon DSLR to Sony A7 MK I then to A7RII for similar reasons you are stating. I had a then young baby and so carrying a DSLR on top of the rest of the gear for a little one was just an epic fail so the camera just orbited the house - pointless.
A move to mirrorless initially meant more use but then in honesty, I recalibrated my "PITA to carry" threshold and then the Sony was a burden too. Its not the body, its the lenses and the rest of the crap. Your mileage will vary. This is just the ramblings of an amateur. I am not experienced with the MK III cameras but they are better than mine were. Mine werent ideal for fast motion. They were workable but they required a bit more effort as the auto focus wasnt quite fast enough combined with my technique, to capture mountain biking for example. This is a reflection on me not the camera. I know people who have shot motorsports easily.
I have since moved from Sony to Fuji and cannot be happier as the lenses are cheaper and smaller. The Sony lenses are just as big and heavy as the Nikon lenses are (laws of physics) so you are really only saving on the body so consider this. The body is a lot smaller and the Sony flash system is superb now as I hear from a trusted source.
Pixel count, will you really notice? You will have less crop margin of error but other than that IQwise, you should not suffer. With the A7III your hard drive will breathe a sigh of relief as you are not filling it with those huge files so quickly.
500-700 shots in a day on a MK I or II A7 series camera would be two or three batteries worth. On the new ones, maybe two batteries.
The A7 III seems to be more gadget laden than the A7RIII from what I have read. It seems to have got some of the nicer features from the A9. It has really good low light capability too apparently.
The A7RII was phenomenal for IQ. The A7 was no slouch
The viewfinder in the A7RII, while an upgrade over the one in the A7, was rubbish for manual focus with legacy glass. If you intend to do this then try a lens out on one before buying.
Lenses. As a rule I know of only one dud Sony lens. The Zeiss 24-70/F4 - avoid. The kit lens is also a bit underwhelming but if you are coming from the 850 then I doubt you are in the world of kit lenses.
The lenses are still big. The 55mm f1.8 which is an AMAZING lens is bigger than the equivalent DSLR version.
The Sony 24-70 f4 is bigger than the Canon 70-200 f4 USM. Neither of which are massively smaller than the f2.8 though they are lighter.
There is only one native 200mm+ option for the Sony. The 100-400 and it is £2,500.
There are a world of lenses there for the Sony now from Zeiss and other third parties so there is no shortage of lenses. Gone are the days of being forced to use an adapted lens to get a focal length that you need.
The price of the Sony lenses is creeping up. They are the same price range as Nikon glass but I am yet to see if they are holding value like Nikon glass.
Sony WILL release a new one by early next year. They like to get people to buy new ones frequently.
The Sony is not sealed like the D850. I never trusted my A7RII in the rain. Lots of stories of water damaged A7RIIs. the MK III is better but still not FUJI sealed.
The image colours OOC are very flat so they need a lot of post processing as a rule.
Having said all this, the Sony takes amazing images! there is a reason why people are moving to them in droves.
Just my ramblings.... hope it helps.
Read this thread too... similar conversation...
https://forums.macrumors.com/threads/thinking-of-going-for-a-m43-system.2113781/