Why in the winter in the northern hemisphere could I not see the moon, even on clear nights, close to the equator?

Your statement is not true. It depends on what phase the moon is in - in the winter a full moon will always be above the equator, whereas a new moon will be below the equator. At first and 3rd quarter the moon is approximately on the equator.

The moon is always close to the ecliptic, which is the path the sun follows through the sky over the course of the year. Think of the ecliptic as basically being the plane of the solar system. The earth is tilted 23-1/2 degrees with respect to the ecliptic - therfore the equator is tilted with respect to the ecliptic. The ecliptic crosses the equator at two points - when the moon is at those points (twice in each 29-day lunar cycle) it will be at (or close to) the equator.