I apologize if some of what I am about to say seems redundant, since it appears you already know how to get Watch Styler working on your Gear Neo. But the problems you are describing with the Gear S appear similar to some of the confusion I was experiencing when trying to use Watch Styler for the first time and went to load new themes. Part of the solution came when I read another post on this forum which mentioned the Gear Faces website as a source of premade themes. It's a great resource if, like me, you lack the artistic touch.
Most of my confusion arose because of how the Gear Manager and the Watch Styler apps work with each other and it took me a bit to figure out what worked by trial and error. It's confusing because it seems the Home Screen Styler within the Gear Manager app will trigger a part of the Watch Styler app when you click the settings icon, but it doesn't actually open the entire Watch Styler app to give you full functionality. You actually have to open the Watch Styler app from the app drawer to get the full functionality, including loading new themes. I'm no programmer so I don't know this is what's actually happening, but it's how it appears to be working based on my observations.
Here's the steps I used: (skip #1-3 if you just want the stock ones)
1. Go to the Gear Faces website and download any WS themes that catch your fancy.
2. Copy the downloaded files to your phone.
3. Open the Watch Styler app and use the Load Themes button to access and load the downloaded themes into the WS app. Since I downloaded the files to my phone, I used the My Files option. But you could probably use the Wi-Fi Direct or Beam Share options to combine steps 2 and 3 to get the files directly from your computer or another phone.
4. In the Watch Styler app, click on the desired theme to select it. Or use the New Theme button to create your own from scratch.
5. Once selected in the Watch Styler app, you can click Apply to put that theme in the list of available ones to use from within the Gear Manager settings for Watch Styler. You also have the option to Edit, Preview, Share or Delete the theme at this point.
6. Go to Gear Manager, then Home Screen Styler, then Clock, select WatchStylerGear3, then click the Settings icon.
7. The Settings screen that appears is very similar to the Theme screen in the Watch Styler app and can confuse you into thinking you are in the Watch Styler app where things should be editable. But you are not, it is only a selection point for themes previously sent from the Watch Styler app to the Gear Manager app. If you just loaded a different theme in Step #5 from what was previously loaded, you will see both the currently applied theme and the theme that "will be" applied. Otherwise, this will be the point where you can select a different theme to use for the "will be" applied selection.
8. Now another confusing step, that "will be" applied means just that: it "will be" applied the next time you switch to the Watch Styler as your Clock Type on the Gear S itself. It DOES NOT change your currently applied Watch Styler theme if you are currently using Watch Styler as your current Clock Type. To get the "will be" applied theme to load, you must change the Clock Type on the Gear S to a different Clock Type, let it change, then change it back to the Watch Styler. At this point, the "will be" applied theme will finally load from the phone to the Gear S and be usable.
I apologize if I have been even more confusing. I know I can tend to include too much detail in my instructions for others. But I am hoping you can follow along well enough to do the switching back and forth between the Gear Manager app, the Watch Styler app, and the Gear S itself to successfully get new themes loaded. Even after doing it several times, I still tend to lose track of where I am in the process. But the big picture way I think of it is this: Use the Watch Styler app alone to make changes to the styles you can use. The Gear Manager app can only see the finished product and deliver them, one at a time, to the Gear S. And finally, the Gear S only can store one Watch Styler theme at any time and asks the Gear Manager app for a new theme only when the Clock Type is changed from a different clock type to the Watch Styler.