I did a bit more searching around and I can't really find anything suitable other than what I've already mentioned. Or something similar, such as MythTV with VLC at the encoding end.

I'e also confirmed that with SageTV you'll need their Placeshifter software to do plug-and-play streaming, because it provides a local UI to the Sage server and its library contents. Alternatively, you can use VLC on the client side if you also install VLC on the capture side. This also requires installing a WebUI for SageTV (written by a third party but works quite well itself - mine is running 24/7, though not doing streaming duties).