Of course I have been told that Apollo 13 (service module out of order) with pure luck...
With luck? Where do you think it was originally headed, Mars?
managed to steer close to the Moon (requiring fuel)
Most of the Apollo spacecraft were already aimed in exactly this way. However, on this forum we like an attention to detail and honesty, and thus we point out that Apollo 13 was one of the missions that deviated from that template. But not by very much. As a first approximation, the existing flight path already carried it around the Moon and back towards home. And this is a natural course for any spacecraft you send to the Moon, whether it is intended to flyby or go into orbit around it.
using the LM engine/fuel/steering aids,
How can you discuss the maneuvers if you don't even know which systems were used, what they were called, and what their capacity was? Did it cross your mind that the LM, for one instance, is an independent spacecraft with the maneuverability to land on another world? You would think that would imply the ability to make attitude adjustments while in free space!
so it could swing around the Moon using its gravity and then,
You describe this as if it was a rare event. Oh, I forgot. You don't believe in planetary probes, either.
at the right moment managed to change direction towards Earth (requiring more fuel), etc, etc, blah, blah, to land safely on Earth.
As an approximation, no. If you entered the approach around the Moon at the right spot, you head back towards Earth automatically. The only "more fuel" is mid-course corrections -- adjustments made so you don't need to make that lunar pass at millimetric precision.
As a thought experiment, what if we remove the Moon? Send a spacecraft out on a trajectory that intersects the Moon's orbit. Don't do anything else. Don't even go near the Moon. Where does that spacecraft end up after a roughly equal amount of time?
All nonsense of course! The NASA SF writers produced a little drama ... assisted by Hollywood. I assume you are sorry you cannot win 1 million Euro?