Might it be that a a coccyx injury be the cause of lower back pain?

Coccyx injury: Yes, the injuries can cause low back pain.

Yes, but not usually: Usually pain in the tailbone is from a degenerated l5-s1 intervertebral disc rather than the coccyx itself. A fall on one's backside can injure the coccyx though and this can be very painful.

Coccydynia: Tailbone pain can occur with or without injury. Not likely to cause pain higher up, though. Back problems, however, may cause tailbone pain. Treatment includes nsaid's, and a sitting pillow. Not a doughnut, but a foam wedge with triangle cutout. This keeps you sitting leaning forward without pressure on the coccyx. If persistent, removal is rarely needed but successful.