Try OSMC [1], it's a massive improvement over Raspbmc and is being developed actively. Works like a charm. If I'm correct, the main Raspbmc dev went over to OSMC a while back.
[1] https://osmc.tv
I might upgrade to a 2B though, it's probably a lot faster (the current setup tends to freeze a little on rapidly changing frames).
Both of mine are perfectly stable at that speed.
I have ATSC (North American Over-The-Air TV) which is similar to DVB-T. I use an HDHomerun Network TV tuner. I watch using my Pi running OSMC and an Add-on for DVBLink, and when I select record, I use DVBLink software which runs on my Synology NAS. DVBLink however is paid software that can run on the Pi, as well as many other platforms (Windows, Linux, Asustor, Netgear, QNAP, Synology, Western Digital).
I tried using TVHeadend but found the interface and support of DVBLink to be better.
My other Pi currently runs Volumio (https://volumio.org) for music streaming.