"A ship in harbor is safe, but that is not what ships are built for."
You said in your question:
It appears that some phases of flight of a piston single, there are sections of the flight for which safe recovery from an engine failure is not possible due to the lack of reachable emergency landing areas.
This is true not only of piston singles, but also of anything that flies out of gliding distance of good landing sites - think about Amelia Earhart's Lockheed Electra.
You'll usually hear this called "inherent risk", but the correct term for the concept that you're describing is systemic risk. As you suggested in your question, this risk has to be taken, it cannot be eliminated. The only way to eliminate it is to not fly; and sometimes, when the risk of the flight becomes too great, that's exactly the decision that a pilot has to make.
Is it possible to reduce this risk with either better pre-flight
checks and through purely technological means?
Yes, as long as you're willing to pay for it. Everything in aviation is a tradeoff, and it's usually a trade either between safety and cost or safety and weight. As an exaggerated example, you could build a piston single with 30mm armor plating to mitigate the risk of incoming artillery fire, but it wouldn't fly very well and not many piston singles routinely take artillery fire anyway.
Those safety improvements which are simple and economical to make, we've already made mandatory (for example, all IFR aircraft certified in the USA need to have an alternate source of power for their artificial horizon if the primary source of power is electric). Safety improvements which aren't simple or economical are left to the discretion of the owner of the aircraft.
Suppose I am taking flight instruction can I do anything to reduce the risk of engine failure?
- Pay attention to your instructor and do what she tells you. You can't prevent an engine failing because an oil pump blows out. You can prevent the engine failing because you ran the fuel tanks dry.
- Fly well-maintained aircraft. Talk to the mechanics who work on your airplane. Do what they say. Avoid operators who let maintenance slide. If you're the owner, don't let maintenance slide.
- Fly aircraft that fly a lot. The worst thing for an aircraft engine is to sit in the hangar for three months, fly for an hour, and then go back into the hangar for three more months. The more that engine gets used, the happier it is.
- Have a contingency plan. For example, examine the area at the departure end of the runway and make a plan of where you would land in the event of an engine failure at any point before a turnback would be feasible. - Thanks Jonathan Walters!
- If you do go down in a field, have a 24-hour survival/medical kit on board, and wear clothes suitable for the climate. Rescue doesn't get there instantly. Speaking of rescue, you did file a flight plan, didn't you?