Pushups are basically planks where you vary the intensity (with the angle of your body to the ground).

Planks build core strength and the lower back is part of your core.

They key is the core(abdominal) strengthening. A strong core is essential to everything being in shape.

If you actively try to engage your core (abs, back, gluts) isometrically while doing pushups you gain more than just arms, chest and shoulder workout. It's about the form - you have to consciously focus on it.

Developed chest and shoulders give you better posture, which is usually enough to alleviate minor to moderate back pains.

I would add squats to the pushup routine (even unweighted). They will develop the quads and glutes somewhat.