An update to my original post concerning random rebooting of the HTC Desire HD.
The answer turned out to be that the SIM card was gradually loosening from its slot, which I think was the original root cause of the phone rebooting unexpectedly (I thought this was something to do with the Alarm Clock, but it seems that was more of a symptom than a cause).
I taped my SIM card down inside the phone so that it couldn't budge from its slot, and it has never rebooted itself again.
That's the good news. The bad news is that the phone is badly playing up these days. When I call other people it randomly puts them on hold and dials other people's numbers, and more often than not it will just terminate the call without warning.
So I think it's almost time to lay the ol' HTC Desire HD down to rest. It will be interesting to see what comes of the Apple, Nokia (not so much) and other phone launches.
When they announce the iPhone 5 (seems like it will actually happen this time!), I hope that the new version of iOS includes proper 'Android-esque' widget functionality - for me this will make or break my decision to buy an iPhone.
So come on Apple, make your OS properly content-centric - not just a bunch of brightly coloured icons.