When we raise the landing gear as soon as safely possible, this enhances our chances of survival in case of engine failure.

In several ways:

First of all, it is much easier to build up airspeed with the landing gear retracted.

Secondly, on several multi engine piston airplanes, the landing gear is operated hydraulically with pressure provided by engine driven hydraulic pumps. So with both engines running, the gear retraction will be much quicker.
Since we don’t know, when an engine suddenly fails, it is reasonable to assume that it is more likely with every second it has run.

Thirdly, at the point where a safe straight ahead landing can no longer be made on the remaining runway, we will be in a higher energy state (faster and probably higher) thus, increasing the likelihood of a safe continued flight on one engine. 

Let’s look at some common scenarios to get a clearer picture:

1. Long runway, VMC.
Here, you would be able to land straight ahead until you cross above the “idle descent flightpath” that would bring you back to the end of the runway.
The more energy in terms of speed and / or altitude you have when crossing above this line (meaning when committed to continue) , the better your chances to maneuver around or to climb above any obstacle in your way.

With the gear down until you run out of the option to continue, you would be much lower, further down the runway and slower, compared to the gear up early case. 

And if you have the gear up and elect to land straight ahead, the chances of survival of a controlled belly landing are higher than of being forced to continue takeoff with a low energy state and maybe even the gear still in transit.

Now think about the same situation but with a cloud deck in 200 feet and obstacles that must be cleared on the departure path.
As soon as the aircraft enters the cloud,  you are committed to continue the takeoff. Would it not be much better to enter the clouds earlier in terms of distance remaining to the obstacles, and in a configuration that enables continued climb? 

I think you get the point.

How about the single engine airplanes?
Well, would you rather have more or less speed and altitude when you have to quickly find a place to emergency-land on a field beyond the airport runway?
Would you rather have more or less speed when you quickly have to transition from climbing to gliding?

Ok, so we agree. 

Fly safely!
Christian Weidner