Was talking to my flight instructor about twin engine planes. He was adamant that they were LESS safe than single engines because 1) you are twice as likely to have an engine failure and 2) the statistics indicate (non-intuitively) that an engine-out in a twin is more likely to lead to fatalities. He said the insurance companies used to think twins were safer but they have adjusted their rates after looking at actual data. I was fully on-board with point #1, but #2 was a surprise to me. I suppose that's what Burt Rutan tried to address with the Boomerang design.
For my commercial MEL training, most of the time was spent on assymetric thrust handling due to a single powerplant failure. When both are working, or neither are working, the plane handles about the same as a single engine.
When one engine fails, while you've lost 50% horsepower, you've lost more like 90% of excess horsepower which is what you need to accelerate and to climb. And assymmetric thrust is dangerous at slow airspeed. You must react exactly correctly and quickly. Failure on takeoff roll means an immediate abort. Failure at takeoff with runway remaining means cutting power and landing on what remains. Failure once airborne and you must quickly identify the failed engine, apply full opposite rudder (toward the good engine), if you don't do this while in a full power low airspeed climb the airplane will absolutely roll over. Many accidents are the result of continuing when they should have aborted, and being anemic or slow with corrective rudder.
Turbine powered airplanes are way different, they have a lot more excess thrust available so powerplant failure is not dire. They have so much more power that they will still easily accelerate to a safe single engine takeoff speed and climb away for a circle to land, which is safer than aborting after certain speeds that smaller twins won't get to on takeoff. That's all part of the computation for "balanced field length".
Or "the plane is certified such that the remaining engine has enough power to take you to the crash site". Indeed.
As outlined above, you double the probability that a mechanical problem occurs. (For p small, 1-(1-p)^2 is around 2p).
Now, if the pilots are very much on top of things (well trained and current), then they can control things and safely land at the nearest suitable airport. However, if they make a mistake (and that's surprisingly easy [1]), it could easily be fatal.
A single-engine plane with an engine failure, in contrast, becomes a bad glider, with a decent chance of surviving the ensuing forced landing.
[1] For example, the TransAsia ATR 72 twin turbo prop clipping a bridge in Taiwan after the crew erroneously shut down the "good" engine.
Just want to point out that "shutting down the 'good' engine" isn't some isolated incident - that was a contributing factor to the British Midland Flight 92 crash [0] as well.
To your second point, my assumption as a non-pilot is that a symmetrically balanced glide is more controllable than lopsided thrust that could put you into a spin?
Not really. Twin engine aircraft don't have to meet the same crash worthiness standards and usually have higher stall speeds (in this case minimum crash speeds).
To be fair the statistics are swayed slightly by a number of factors, training accidents and different risk profiles of the flights.
FWIW: 5 years ago or so I was on a plane that lost[0] an engine in rough weather shortly after take off.
The only reason I know is because the captain informed us about it and added that since it was technically an emergency landing we would be greeted by the fire department.
He then turned the plane around and landed as if nothing had happened.
The bloke across the corridor, an old pilot, complained about how there never happened anything remotely exiting on any of his flights ;-)
[0]: to be as accurate as possible I think what he said was they'd lost oil pressure on one engine and decided to shut it down voluntarily.
