Joints when mobile act as buffers to reduce forces that travel up and down the skeleton. Where joints are immobile , these forces may contribute to pain causing mechanisms removed from the site of the joint, by allowing torques and adventitious movements to become severe. TheSIJ pair are a case in point. Without the force reduction effects of this pair of massive slow moving limited joints, forces travelling up the legs and pelvis , may not dissipate normally and continue on into the spine.
I personally am of the view that allowing a pair of apparently immobile joints that may impact on spinal behaviour to go unchallenged would be improper in a well designed treatment.
I do spend time to restore SIJ movements , which occurs almost all the time in those with overpronation problems of the feet, and find long term benefits from doing so , in concert with a range of manual therapies. The test I use is known as the STORK, which I have always found to be satisfactory. It is a movement test of one lower limb at a time and , in my view , is sufficient to demonstrate the mobility , or lack thereof of SIJ's. My results are usually very good, in those where this loss of movement is the consequence of altered biomechanics, ie, pronated feet. As stated earlier those with immobile SIJ's do not present with SIJ pain , but may have referred pain, to the sacral area , from lumbar structures associated with the disadvantage of this loss of movement.
While your interest in the posture and position of the pelvis is fascinating neuro, and it would seem consumes your energy and time, I would prefer , given there seems no advantage for me or my patients to alter my aproach, to just offer my congratulations on a stirling effort and my best wishes for further results . I hope your time is well spent.
Cheers