When using a separate 'smart' TV box (streamer/media player) it (usually) connects to the internet via wifi or Ethernet.
Q: When connected to a router (wifi or cable) can the device then see a NAS on the network, or is it internet access only?
If you have your IP addressing set up for a subnet, the media box can see the other machine(s).
I have a raspberry pi with kodi and it can get files off my PC , as well as get things off teh interweb.
I just configure kodi with the IP address to use for my PC and off it goes.
I do something similar with a Raspberry Pi running Rasplex. It sees the NAS just fine on the LAN.
I encourage you to move in that direction.
The client/server model, with media stored on a NAS/server and networked streamers is a really good one. Set up right, this type of arrangement can be very easy to use, with nice eye candy. Plus, you no longer need to remember which room has the media player with the file you want on it (Like I use to have to do with hard drives connected to my old WD Lives). My setup has a pass-mark against the most demanding of benchmarks - the Wife Acceptance Test.