Science-is-Truth

Sorry but your question makes NO SENSE and is impossible to answer. The REMOTE control DOES NOT USE WIFI OR ETHERNET, it uses RF direct from the remote to the TV. So you were able to ask the TV, via the remote control, to download something from the internet, remember simply making the request DOES NOT REQUIRE ANY NETWORK INTERACTION. However the TV now needs a network connection in order to get the APP from the Internet. Now that connection can be WiFi or Ethernet. Are you sure that the TV uses Blue tooth to connect to its remote control, it is more usual to use Infra Red, which your phone probably does not have. You have to construct your questions in a more concise manner, otherwise anyone trying to help you is simply GUESSING at what you are talking about. For instance you use the phrase "one TV's button" DUH?

Upgrading to Gig Ethernet will have ZERO effect, in any case there is NO WAY to add a different Ethernet adapter to an existing TV. Its a TV not a PC, on a PC your have the freedom to pretty much whatever you wish as long as you have the harware port and software drivers to support the adapter., The problem is between your router and the rest of the Internet. Its like having a private drive that connections to the interstate on which you can drive at 100MPH, but traffic on the Interstate is only going at 50 MPH. Guess what your journey and your data will travel at the slower rate