In my case, the PC that would be used as the server for library management (because it has an optical drive and has a proper desk and chair) is quite beefy, therefore uses a lot of power and it's quite noisy, it's a desktop machine, designed for desktop applications that require processing power. My NAS is a very low power and quiet device, and is left on all the time as it has several central services running for servicing several devices (e.g. file serving, CALDAV server, web server etc). It could also act as a media server (DLNA) but I use MC instead for its flexibility in cataloguing and indexing. My "HTPC", the living room PC where all media is played, is low power and silent with a mobile-level processor, it uses a max 26W at full load (not very often), about 11W when playing music via MC, and it just about manages ROHQ with some tweaks to MadVR. My NAS (currently a 2-bay unit with 2x 3.5" 4TB physical spinning drives in it) is about 19W at full load, 10W when idling. Needless to say, my desktop PC probably uses more power than both combined. So Yes, to answer your question, NAS's generally do use significantly less power than an average PC, especially when you need two PCs switched on to use MC. They have considerably less powerful processors and the fact that they run Linux helps to keeps the cost down and allows a less powerful machine than Windows would.
In order to play just one MP3 file, I would therefore need 2 complete PCs and a NAS switched on. This is not very good. Client/Server technology exists, for one reason, to make processing more efficient. Put processing power where it's needed, send processing the right place, use the right tool for the right job, use a PC with a beefy processor for number-crunching, a low-power machine for menial everyday continuous tasks, a device with decent graphics and audio for media playback.
Many, many people have requested a NAS-installable server version (look at the "Would you buy a Linux version" thread for a sample), it crops up quite regularly (see this thread, this is what the OP is basically getting at - his PCs are not left on all the time - "My concern with having a single "server" or master version of MC that needs to always be running is that the most logical source for such a server would be a HTPC or some other computer that is always on and not doing much else other than serving MC."), therefore demand is there and there's obviously a case for it. It makes a huge amount of sense, it's a server so put an MC server on it. Leave the client bits to the clients.
It doesn't require much processing power to serve files and service database requests (in fact, it's what NASs are designed to do) therefore there is a huge argument against leaving a beefy PC switched on all the time just to do this.
I take your point about Minnesota - we have the same problem here in Wales.