...based on your principles there would be no SE flight, other than a few trivial routes.
That's really not true! Coming back to the original question, observing a reasonable minimum ceiling you can take the factor "luck" almost completely out of the equation. Where I fly SE IFR, this value is 1000ft, and it is not flat, uninhabited terrain, but there are mountainous regions like the Black Forest and also some larger cities and towns instead!
I don't think these are self erecting i.e. if you fly in a 30 degree coordinated turn for a bit, the "gyro" will indicate a level roll attitude.
Yes, of course. But that is not what you will do in a limited-panel emergency situation. All you need is to keep you wings level and work with small changes in pitch and direction from there on. Just like you would when flying with turn-coordinator, magnetic compass and altimeter.
What app do you use?
Myself none, because I don't have an iPhone or iPad (although I've been a Macintosh user for over 20 years). But all our students and co-pilots play with these things all the time, so I feel very safe in the event of an instrument failure
Here in the UK, there is rarely any GSM connectivity above 1000-2000ft.
Really? During single-engine flying (rarely above FL80 in my case), I usually leave the mobile phones on and have a connetion most of the time.
It ought to work in an emergency, but a handheld radio is a whole lot easier.
Sure. But every pilot has a mobile phone and very few have a handheld radio. And even those with a radio - when did they last charge their battery?
...but the controllers are not on the phone (on public numbers).
Many tower controllers are, especially at smaller (but controlled) airports! We need to call them all the time to co-ordinate our training approaches. And if you have the first one on the phone, he will be able to either connect you to the next one or at least to relay your message. Talking to ATC on the phone even briefly with a bad connection in case of communication failure will take most of the stress away from the emergency.