I've had three engine failures very low after takeoff. One, a fuel interruption, put me back on the remaining runway (I could not have made it past the end
), another, at only a couple of hundred feet was ice crystals in the fuel, into the next field, no damage, and took off later when resolved. The last was the mouse nest of insulation from under the glare shield, being sucked from the air hose, into the venturi, with the application of carb heat, again, into a field, no damage, towed it home.
Very much worse has occurred during testing I was doing, but I was up high, and testing for it - it happened! Consider the tried and true C 180. Owner installs a STOL kit - yippee! Look at those high AoA's and steep departures it can do now (not safely). Then owner installs a larger carburetted engine - still no fuel pumps. Wow, it goes up even better now! But.....
When you combine the higher pitch attitude capability with a STOL kit, with the higher fuel flow demands of a bigger engine, you can get the plane into a very nose high danger corner, in which it will quit, and from which recovery to gliding flight is unlikely. The carb inlet is actually higher than the tank outlet - 15 degrees pitch up is in the danger territory. I agree, this is far away from the regular airport circuit technique, but there are pilots doing, it and I have done the testing to demonstrate that there is a much greater risk of EFATO doing this. Floatplanes are more vulnerable, as pilots are sometimes trying to outclimb terrain. A friend of mine, a very competent, though perhaps a bit complacent, friend of mine broke his back in the resulting crash - and he has heard my admonishments on this (beforehand).
So EFATO's are happily rare, and there are things you can do to make them even less likely. But, flying with "room" for them to happen without ruining your day, is a good idea when you can.