I use an R5 and find it a great "Workhorse" of a NAS case. Definitely recommend putting an extra fan in the front for 2 x intakes. I also converted the top 51/4 bays so they now hold 2 x HDDs so the system is now running 10 x 3.5" HDD.
Not a single problem with cooling. Plenty of room inside for airflow too. The R5 being primary built to suppress noise does a great job and is still able to maintain an acceptable temp, even with 10 HDD's. I found the stock exhaust fan to be more than adequate.
Personally, if building a NAS, you ideally want that to be its primary task, I would not be wanting to use such a system as a regular client machine for various reasons.
The reason I mention a second rig for use under the TV is that although many prefer to use TV Apps and smart sticks for accessing services and Plex etc, there is nearly always a compromise. I found running a full blown, albeit nothing too fancy PC, there was little compromise, albeit no remote control but that keyboard/touchpad helps in that regard. I have Parsec installed so use it for the more casual games streaming from the main PC for couch co-op etc.
As you suggest though, yes, the NAS hosts the content and other devices simply connect to it.
Regarding the PC specs for the client on the TV, I built mine late spring, before prices went nuts. You cannot source anything for its actual worth price at the moment so not much help there I am afraid.
I just went with the 3200G as it had an iGPU which was capable, MATX case (fits nicely with the TV) and not as cramped as a ITX build which is also more expensive. When looking for MATX boards, look for 4 DIMM slots to give yourself an easier upgrade path and also a few PCIe slots just incase. Some MATX boards at the lower price points are BARE.. some only have 1 chassis fan header which is pants. For the most part though, you do not need anything flash, standard MATX case of your choosing, Standard MATX board, 8GB RAM (If your just using Windows and watching stuff from the server, 8GB is fine to start with) and a SSD of your choosing. The system is mostly using soo little power, your cannot hear it, I would say it is almost inaudible. Get the PSU AFTER you have chose your case as some MATX cases use none ATX PSU's. Personally, go with a case which supports ATX PSU's, it just makes things easier and cheaper
The 3400G has a much better iGPU but for me, if I want to play games on it, I would just put the money towards a GPU.. but that was not the purpose of the build anyway.
Alternatively, get a ROKU box but what is the fun in that