If it's any concern, the 2012 Mini idles at 16W, direct streams at 25W, and jumps to 50W max at full load. I also used to use a NAS as storage backend (via iSCSI, before I had the mini) but rapidly found out that it's quickly IO/bandwidth constrained as soon as 3 devices direct stream high bitrate 1080p content off of it, I've recycled it as a backup storage now. In these cases, it depends on the output resolution, but usually 2 or 3 transcoded streams are easily handled by the CPU. I need to reencode only when streaming over WAN/3G (the joys of DSL.), some cases of wonky subtitles that have to be hardcoded before playback on iOS, or when the audio tracks are in FLAC (to iOS as well). I can direct stream without re-encoding to most iOS devices provided they're on wifi, and simultaneously to my other devices over wired GB LAN (a 2011 mini serving as HTPC and my ISP's triple play media player over DLNA) I'm using a 2012 quad core Mac Mini Server exactly for that purpose, but instead of a NAS, I have an external Thunderbolt Raid-6 enclosure by areca. You could probably dig up a smaller case too but I wanted one that'd let the CPU draw air directly from the outside. So basically, it comes down to wanting to trade the extra 1-2 streams for a case say 4x the size. In this situation, none of those should matter. The only advantage the NUC has is it's smaller footprint, the IR sensor, and built in wireless.

Without that requirement, you could definitely build an mitx system that smoked a NUC for same or cheaper (the 4590 I was looking at has a 7242 passmark). My requirements also called for a motherboard with vPro which drastically cut down selection and increased price. It's quite a bit bigger but still pretty small and had a much faster proc and dual NICs. A DIY system built around a mitx board and an Antec ISK110 was just a hair higher than a NUC when you add the RAM and storage to the NUC. However I recently was comparing a NUC to a DIY system for my home lab. So 2 streams would be fine, 3 maybe if 720p.

The i7-5557U which comes in the NUC I mentioned has 4760 passmark score. Very roughly speaking, for a single full-transcode of a video, the following PassMark scores are a good guideline for a requirement:

Plex media server hardware specs