The case seems to just be plastic and the PCB inside seemed fine, so I put the two pieces of case back together and taped them back together. I went out again and after about a minute of flying (on a new battery) the quad started diving left and down and then seemed to recover a bit and then down and left again before crashing and breaking another prop. I put it down to just a rookie mistake (after all, it was only my second crash).
I took it back home, recalibrated all the sensors (as the first crash was 200 miles away where I was on holiday) and took it out, again with a fresh set of batteries. Almost the exact same thing happened, which got me thinking that maybe the antenna was actually broken. I spent most of the day trying out different settings on the receiver for loss of signal, but each time they'd work the first time I switched off the receiver and then not again. It turns out that the receiver overrides the loss-of-signal settings in the receiver, so there's no point in following the instructions that came with the recceiver.
To change these settings, on the model setup screen (2/12 on the model), at the bottom in "Failsafe mode", select "Custom" and then "[Set]". These then work consistently over multiple power cycles of both transmitter and receiver. I set my defaults to turn on stabilised mode, turn on the beeper (not that I've got one fitted yet) and throttle at -20, which seems to be just below hover on a fresh set of batteries. I figure that will allow a gentle descent and hopefully let me get closer or a better signal somehow and regain control rather than having it plummeting to the ground. Of course, I guess with the props still going when it lands, it'll probably destroy the props and/or motors in the process but that's probably better than a half kilogram falling uncontrolled from the sky!
And perhaps the root issue was that the battery voltage on the transmitter was down to 7.6V. It feels like it should be OK at that charge, but after topping up the charge AND setting the failsafe I managed about another 30 minutes yesterday without any incident at all.