Perhaps let me clarify two points:
1) I was talking in my previous post about the iOS app 'Apple TV Remote'. It seems mostly unusable for my setups where connections are ethernet cable based. The app never fails to show my Apple TVs, but is most of the time unable to connect to the Apple TV it lists. The hints offered such as go closer to the device and make sure both devices are connected to the same WiFi, while my Apple TVs are never on the same WiFi as only cable connected, all this makes simply no sense. So Apple should fix this iOS app for its buggy behavior or start deprecating it.
2) Fortunately with newer iOS there exists an alternative that seems to work reliably: This technique, described here Set up the Apple TV Remote on your iPhone, iPad, or iPod touch - Apple Support, seems today to be much better supported by Apple and does not require aforementioned iOS app 'Apple TV Remote'. While this seems to work in my experience reliably also with a setup involving ethernet cabling, it also gives misleading information. E.g. in case of some difficulties also this system software consistently refers to the misleading need that both devices would have to be connected to the same WiFi instead of stating the same LAN. Any user using an ethernet cable is invited to disconnect her Apple TV first from a perfectly working ethernet cable and try to use the WiFi, while the actual problem has nothing to do with the fact that an ethernet cable connection is involved.
To wrap up: Fortunately the newer iOS offers a solution to this problem. Yet, buggy software becomes more and more time consuming for all end users and Apple's software quality is unfortunately also only declining. As a programmer I am well aware that at the same time the complexity of the software has exploded, so one can argue that relatively seen the quality has even increased. However in absolute terms end users are suffering more as our societies rely increasingly on software. Therefore I find companies such as Apple and others need to accept that if a software behaves as poorly as the iOS 'Apple TV Remote', this is simply not acceptable from a consumer's perspective.