Seventh chords are the very beginning of what we call "color chords." They naturally lead us into the next chord in the the chord progression. You certainly can play a song without sevenths (most of the time), but a song loses its vibrancy and color. Learning 7th chords also leads the way to understanding augmented chords, sustained chords, 9th chords, 13th chords, etc.
Consider that music is another method of communication, just like reading or speaking. When you learn to read, at first you are fine with "See Jane run" and other such simplicity. But as your reading progresses, you develop a much better picture when you read, "In her bright red athletic top, Jane sprinted down the path as though the hounds of hell were after her."
Simple chords with their root, 3rd, and 5th are great. They are fine on their own. However, when you add what we call "color chords" to them (7ths, 9ths, 2nds, 13ths, augmented, sustained, etc) you begin to create mood and meaning that a simple melody with a basic chord cannot convey. Don't give up! You'll begin to hear the difference and see how it adds to your playing!