I'm currently doing something like this, and I agree with your Synology idea.

Here's what I'm running: Lenovo M900x Tiny i7, 16GB & 512GB SSD. Ubuntu & Docker w/Plex, CouchPotato, Sonarr, Headphones, and much more in Docker containers.

Synology DS1511+ with external five-bay expansion running about 24TB usable. I just don't have the time to mess with a self-supported NAS any more work keeps me too busy. I'm not saying no to a FreeNAS or other software based NAS but if you have the time and energy, go for it.

I use Plex for my portable devices, and I use KODI/LibreElec at the Televisions. It gives me, IMHO, the best of all worlds. I don't need to transcode in my house; I can also download subtitles on a whim at the TV when I want to watch quietly and not disturb the sleeping wife.

Three TV's: * LG 4K UHD - Plex & Netflix also has a Lenovo M900 Tiny i5 running Kodi/LibreElec @ 4K (overkill but I happen to have one) * 42" 1080p w/RasPlex and LibreElec (two RPi's) * 40" with an ancient Acer Revo Ion that refuses to die running LibreElec

If I had to do it all over again and start like you I'd look at:

1) Synology DS1511 non-plus. You're not going to have my NAS to the transcoding, let the NAS be a NAS. Future expansion big bonus. Also, Synology's tech support is simply awesome.

2) The most powerful machine I could afford to run it for the foreseeable future as a server/transcoding engine.

3) What would I do at the TV for 4K support right now? * Perhaps one of the 4K Roku's * Nvidia Shield * PLEX client built into the TV (Didn't know I had this until I'd bought the TV first, had purchased the Roku 4K Ultra on a pre-Black Friday deal, haven't touched it since) * a Kaby Lake NUC type system that does the graphics running a variant of KODI * Hold out for a 4K supporting Raspberry Pi with RasPlex? (a good option for a non-4K tv too.)