I've owned several Land Rovers over the years with three different style tailgates. My original Discovery and Discovery II both had a swing door, my LR3 had a split tailgate, and my current Range Rover Sport has a single-piece tailgate. Of the three, I prefer the split-tailgate for several reasons:
When you park facing up a hill and then open the tailgate, the lower section of the tailgate will keep items from falling out onto the ground. (I have dumped luggage, groceries, golf balls, etc. onto the ground many times after they shifted position while driving.)
A split tailgate gives you a clean surface for leaning against or sitting on while changing shoes, drinking, eating, etc. (A single-piece tailgate means that you sit on the external bumper, which can get your clothes dirty and/or wet.)
The lower section of the tailgate in the "up" position usually prevents my dog from jumping out before I'm ready for him to jump out, so I have time after opening the tailgate to attach his leash, make sure other dogs are not around, watch for other cars and people, etc. Then I can release the lower tailgate and let him out when all is clear.
With a split tailgate the upper section is smaller, so it doesn't need as much space to open. This is handy if you are pulling a trailer or have a hitch-mounted bike rack (or something similar) and need to grab something out of the rear cargo area.
Probably not a use case that applies to most, but if you need to shoot photo or video from the car while on the move, you can sit in the cargo space and shoot with the upper half of the tailgate open (no glass to interfere with the image quality) and the lower half closed. The closed lower half of the tailgate provides some safety because it makes it harder to fall out or drop things that then roll out onto the road. Watch Top Gear carefully and you will see them doing this with the Range Rover quite often. (I worked as a professional photographer for many years...)
Thanks for the detailed reply! That really helps. We've never had a vehicle with a split tailgate so I was trying to understand why do other people actually like them (i.e. specific examples like you listed). Your use cases make a lot of sense, but I guess we'll just have to see how it works out for our typical uses (groceries, dogs, and loading bikes). I'm sold on the R1S, given everything else it has going for it, but I'm on the fence about this one feature.