From the version history of Remote S from October 5, 2016:
"Watch app is now working on WatchOS 2+ with speed optimization upgrades from WatchOS 3. This means that it doesn't need to be connected to your phone to work."
You also confirmed that Remote S works with just WiFi and without the iPhone turned on. According to the Apple site, it says "If your watch is connected to a cellular network, but your iPhone isn't nearby, your watch can do the same things that it can when using Wi-Fi." Meaning that there shouldn't be a difference between using WiFi with the iPhone turned off and using the cellular network but your iPhone isn't nearby. So it should have worked for you, and it seems like it works for other people who are only on LTE.
Also, does it work if the Watch is on LTE, far away from the iPhone but the iPhone is turned on and connected to the internet? The article you linked to implied that it should help some apps to work if this is the case.
There's also a "Disconnect from Watch app" setting in the Remote S settings on the iPhone. Perhaps turning that on would allow you to use Remote S untethered from the iPhone?
And finally, there's a written review of Remote S on iTunes from Sept. 25, 2017 that says that he/she was able to use Remote S without the iPhone from the new Apple Watch with LTE. He/she also wrote "Is it possible that the app can keep working if I restart the watch while away from my phone? Currently seems like it's coded where I need to start the Watch app while connected to my phone". Not sure if that helps, but maybe that is the same issue you're having? Try starting the Watch app while connected to your phone first to see if that works. And then perhaps the "Disconnect from Watch app" setting would allow you to use it untethered?