That will work fine for your requirements. You will need either an extender or another PC (like a NUC) for each TV. While a PC would work since the HDHR Prime can dynamically allocate tuners, extenders would be a better solution for DVR purposes. With extenders all recordings would reside on the central PC and any recording could be accessed. When using PCs any recordings that are flagged as "copy once" could only be watched on the PC that was used to record it.
Also, Ceton makes a
6-tuner cablecard tuner. While it is more expensive than the HDHR Prime, take my word for it, once you get used to having 3 tuners you will wish you had more.
There are some caveats and some "bonus features" - depending on overall amount of use and requirements.
First off, perhaps 3 tuners will be more than enough. So far, I've never run into a problem. You have to factor how many TVs will want to watch a live show at any given time, and/or how many recordings (plus live shows being watched) you'd want to be able to handle.
At the time, I felt far more comfortable picking up the HDHR 3-tuner model over the Ceton 6-tuner. At the time (July), many users will still experiencing some stability/driver issues on the external 6-tuner model from Ceton. At the time, the dynamic tuner allocation was also still a work in progress, whereas I had heard it was rather solid on the HDHR models. So I went with that, and have been more than happy since.
It's bound to be improved by now, but... what you really need can only be determined by you.
Now... other things. If you use Windows 8, ONLY the Xbox 360 can be used as an Extender. Someone correct me if I'm wrong, as I don't feel like searching, but I have also heard the Xbox One does not function as an Extender, as ridiculous as that sounds.
There are ZERO third-party Extenders that work with a Windows 8 MCE host.
If you have Windows 7, it's not that much better - there used to be more, but the only actively produced Extender left (aside from the 360) is Ceton's MCE Extender device, and it's not the cheapest device out there (considering its lightweight purpose).
Also, regarding Windows Media Center Edition - unless you have what might be the best cable company to ever exist in the world (I don't think such a company truly exists), you will most likely have channels, or at least the occasional program, flagged "Copy Once".
Most of the "cable" channels, especially AMC, and of course the big movie channels like HBO/Max/Starz, as well as some sports programs, tend to have that flag enabled. Basically, it's DRM. What it means for you, as a CableCARD user - you must use Windows Media Center Edition, or you don't get these fancy programs/channels.
No other media software can touch the DRM content, at all. It can't watch the live show, it can't record it, and it can't even read the files once recorded (assuming you have a valid MCE PC to record).
That's why the concept of Extenders is something Microsoft truly needs to work on, yet has let stagnate, because there is nothing we can do about sharing content around the house, if we use this setup and adopt the latest software and hardware.
Now, let's say you want to watch a show but generally don't want to worry about recording it or having it get buffered on the central HTPC/server. If this particular device never needs to access previously recorded content, any media info/guides, and you don't care about buffering content for rewinding/pausing (I MIGHT be wrong on that last one, don't think so though), you do have the option to use DLNA devices.
I have watched TV, using the HDHR on my network, through my PS3. It basically is direct communication between the HDHR and PS3 on the network, without the HTPC. It does count as an active tuner allocation, so, on a three-tuner device, it's one of three in use.
Another concept: any PC with MCE can latch onto a tuner and watch content, as well as record. It's generally not recommended because it's basically a solo instance - unless it's DRM-free content, you can't watch it on other devices if you record anything. That said, you would have the full MCE experience, in case using pause/rewind/forward is all you want on that particular screen. You can record just fine, if so desired, but you should expect that to be only accessible locally. It would be one of the tuners in the available tuner pool, but ultimately would function as an entirely unique DVR. If you accept that use, it's still pretty damn awesome to be able to pull up your entire cable subscription package on your PC/laptop as long as you're on the same network. I wouldn't advise on a weak wireless connection, but on a good wired connect, it'd be just fine.