Sony promised and has delivered on better JPEG images too, with more true-to-life color accuracy and less aggressive sharpening than before. I tend to shoot both RAW and JPEG images to have a backup, and for one particular photo, I was comfortable using a JPEG in place of a glitched RAW image.
Early reviews of the A7R III indicated that it didn't work with Sony's PlayMemories smartphone app, but Sony seems to have fixed that issue -- it worked fine for me. The app lets you connect to the the camera via NFC, WiFi and Bluetooth, see a live view, change settings like f-stop and aperture, and trigger photo or video capture. You can also transfer images from the memory card to the app. PlayMemories has always been clunky and connecting to the camera can be a pain, as its meager 2.2 iOS app store rating shows. (It has a much better 3.8 score on Google Play.)
With 8-bit (16.8 million colors), Sony's A7R III may not support 10-bit video and a billion colors like Panasonic's GH5, but it's still arguably the best full-frame, mirrorless camera for video on the market. There are two 4K shooting modes: One at full-frame with some line-skipping, and the other that uses APS-C cropping with a full sensor readout. The latter produces a better image if you peep close, but if you want full-frame depth of field, you'll have to settle for slightly more noise and moire. Personally, I could barely see the difference.
Continuous focus worked very well on the A7R III, locking solidly on subjects in decent light. It was less accurate in dim illumination, but the results were still acceptable. Focus speeds when shooting video are considerably faster than before. Suffice to say, it's much better than my A7S, so Sony has made some big improvements over the last three years.
All the 4K video I shot in dim, drizzly conditions at 100 Mbps in standard mode was sharp with rich, accurate colors. If you'd rather sort out colors in post-production, you can shoot in the neutral S-log 2 mode, included with the A7R III for free (Sony used to charge for it). Sony also offers a new shooting mode called Hybrid Log Gamma (HLG) that's also found on the Panasonic GH5. That saves metadata so you can easily create HDR-ready video.
As before, Sony gives video shooters both microphone and headphone inputs, audio levels, histograms and pretty much everything else you need. The five-axis in-body stabilization, coupled with lens stabilization, makes handheld video about as steady as it can possibly be. The extra heft also helps in this regard, I find. Overall, the A7R III is about as close a pro video camcorder as you're going to get in a mirrorless camera.