It does sound as if it needs a retune, or you need to find out how to select it as the "input" to the television set. If you only get the usual 5 stations, I'd suggest you're watching analogue TV. Powering off the thing will stop the analogue signal going through it, so your picture will disappear - as it does.
Assuming there's a SCART connection to the TV from the device, you need to find how to select the SCART input. If the only connection is the conventional aerial cable, it's pretty certain the digital bit isn't connected at all.
These things do tend to run very hot - I have a Topfield PVR (same concept, different brand) which works extremely well, but gets very hot. The hard drive in that seems to spin much of the time, but isn't obtrusively noisy.
It gets so hot that the capacitors in the power supply break down through the heat after a year or so, and have to be replaced (just done that). Next week's job is to fit a fan in there (if I can shoehorn one in) to cool it and avoid another soldering iron job.