12 feeds will require, IMO, either a >=24" monitor, or two monitors, for the feeds to be useful in live view.

A NUC will let you use as big as a monitor you choose, and in my experience will run the feeds nicely if you go with the top of the line.

Any all in one system will limit your monitor size/resolution, unfortunately. Otherwise, I'd suggest to look into those as well.

I am not sure a NUC supports multimonitor, but it will definitely via USB3 at least.

Also, you can attach it to the back of a monitor compatible with VESA arm suppors, saving a lot of desk space.

I am currently using three different NUCs at work, for very different projects, all running Winosws 8.1, and 1 at home as a HTPC, and they all proved extremely reliable in the last year. They are very nice machines.