Should not the notes of the triplet figure in Beethoven's "Moonlight Sonata" be so blended into each other that you do not hear them in separate notes, but as a background, so to speak, for the notes in the melody?

The truth lies midway between two extremes. While the accompaniment should be sufficiently subdued to form, as you say, a harmonic background, it ought, nevertheless, not to be blended to such a degree as to obliterate entirely the undercurrent of a triplet motion. The accumulation of each chord should be produced through the pedal, not through an excessive legato touch.