This is a bit of Tiger Moth old folk lore theory from the days when engines were not so reliable.
If you fly from Bournemeouth to Jersey(which will take a lot longer than a 2 miile final) and have an engine failure you will never make a runway either and given the choice I would rather land in an undershoot than in tbe English Channel anyday!!!
The period of the circuit on which you are on final is only going to be one or two minutes usually and you will be at low power setting. In terms of risk you should me more concerned about the first 500 feet in the climbout with the engine working flat out because you arnt going to be able to make the runway then either, are you? However at an airfield that does have a hostile approach area consideration should be given to flying a full glide approach to the threshold
Excuse me, I really do not want to be rude but I have serious difficulty with all of this.
"Flying over the sea is more dangerous" ? Flying over water (or other areas with little options for a forced landing) is the PiC's option, to be considered before flight. Risk assessment and all that stuff. But the one risk has nothing to see with the other. It is not because one accepts the risk of flying of over possibly dangerous areas that one should take unnecessary risks in the approach or final phase of flight.
"A bit of Tiger Moth folklore" come on, give us facts, or at least arguments, not tabloid talk
"when engines were not so reliable" so a risk that was not acceptable with a 99,0% reliable engine does become acceptable with a 99,9 % reliable engine? Or are you the inventor of the famous long wanted 100,0% reliable engine?
"Risk on final is limited" yes it is. Limited but not zero - unless at a safe altitude. Which is in easy reach for the vast majority of planes and the vast majority of pilots.
"The first instants after take-off are far more dangerous" yes they may well be, depending on the surroundings. But again, that is no excuse for choosing a less safe procedure for final.
I say again because it seems to go down hard with some:
There is not a single reason to NOT come in fairly high, cut power as soon as certainly within gliding distance from the threshold, and sideslip off the excess altitude so as to touch down on the numbers.