I think when I get the new modem i will leave it on 24/7. £7 a year seems small fry. I certainly don't want to trigger an environmental debate. I was more concerned about the modem over heating than the cost of running it 24/7.
For what it's worth, I share your concern over fire hazard it's something I could worry about quite a lot, if I allowed myself to do so. I can remember, as a teen-ager of the 1970s, my father actually switched both gas & electric off at the mains before going to bed, just for peace of mind.
But it (leaving things on 24/7) seems to have become unavoidable in many aspects of the technological age. DECT phones for example, fridges, intruder alarms, the list is endless. My view is that I just have to trust the manufacturer to ensure the risk is very very low, (and make sure my home insurance is up to date!).
Even my car, I recently discovered, springs to life while locked up in the garage in the wee small hours, running its ventilation fans at full speed for 10 minutes to deodorise the system.
And in fairness I do suspect that modern electronics apparatus is several orders of magnitude safer than that of 50 years ago.