Do you have wired ethernet and power available in the Attic? If so I'd consider putting a HDHomeRun Dual up there (assuming it doesn't have extreme temperatures, otherwise on the floor below as close to the antenna as you can get) to minimize the coax run from the antenna. That would give you two tuners. But even if you plan on doing it the way you are thinking about, TVHeadEnd is going to take the signal from the tuners and change them to a digital stream that is then sent via Ethernet, and at your TV's you will need some type of media center box (could be a Raspberry Pi 2 but you might want something with just a bit more CPU power) at each TV, in order to run Kodi or VLC (I'd suggest Kodi). There are a lot of relatively inexpensive Android-based boxes that run Kodi (but I would look for recommendations on the Kodi forum). Of course if the Raspberry Pi performs well enough for you, than you could just get a couple more of those.
Not only will Kodi give you more viewing options (for example if you have video files saved to a hard drive on your network or a local hard drive, Kodi can access and display that, and Kodi also has a repository with several LEGAL addons that will let you view online content from various sources, though in that regard they don't provide the variety of, say, Roku or FireTV). Just be sure you get Kodi direct from the kodi.tv site and any addons from the official Kodi repository; there are a lot of other sites out there that will try to fool you into thinking they are Kodi sites but they aren't, and the software they offer can damage your system and/or get you in trouble with the copyright cops.
VLC will also work for displaying live and recorded content from TVHeadEnd and your tuners, but on a regular TV it's not nearly as easy to use, and on some systems the video output of VLC is much more "jerky" than with Kodi.
So basically the idea is you run coaxial cable (preferably as short as possible, but as long as you get adequate signal strength then the length doesn't really matter) from your antenna to your tuner(s), then (if using a HDHomeRun) the signal travels via Ethernet to your TVHeadEnd backend (otherwise your tuners will likely be installed in your backend system), and then from there via Ethernet to your media center PC's, which are then connected via HDMI cable to your TV's. For alternate viewing options, you can install Kodi on your PC, laptop, or on some tablets. You configure TVHeadEnd via a web-based interface so once you have your backend server set up you do not need to leave a monitor and keyboard connected. It ispossible for your backend server to do double duty as both a backend server and a media center PC (in other words, you have both TVHeadEnd and Kodi installed on the same machine, and you have it connected to one of your TV's). But many of us just prefer to keep our backend servers separate, in a location near where the antenna wires come in, though again I would not put it anywhere there are temperature extremes or high humidity.
One last point, some tuners work better with TVHeadEnd than others, and even with HDHomeRuns, some people find that TVHeadEnd works with them with no additional software installation, while other have reported they've had to install the HDHomeRun Linux software (which brings in a bunch of dependencies, so try without it first) to get them to work. In theory any TV tuner that is recognized by Linux should work (if they only say they support Windows they may not) so before you buy anything you might want to try to find out if others have been successful in getting it to work with TVHeadEnd. We have three HDHomeRun Duals (so, six tuners in all) and they work fine in TVHeadEnd (this all assumes you are in an ATSC country such as the USA or Canada, if not then make sure before you buy any tuner that it supports the local TV format, whatever that might be).