tmarkowitz,
As to bursitis showing up on an MRI, yes it will, if it is extensive, with a significant amount of fluid within the bursa and inflammation around the outside of the bursa. Mild or moderate bursitis may not show up well on a noncontrast study.
When you say that the pain in on the outside of the hips, that is usually due to greater trochanteric bursitis or tendonitis of the tensor fascia lata (this muscle's distal tendon in the ITB, or iliotibial band, at the knee). Anterior hip pain, if below the groin area is also usually due to a tendonitis of the hip flexors or possibly a bursitis around the iliopsoas tendon as it goes over the anterior pelvic brim.
If the pain is in the groin, over the true hip joint (basically where you would take a pulse in the femoral artery), then the pain is probably coming from the joint itself.
The labrum is a cartilage ring that goes around the outside of the acetabulum (hip socket), to deepen the socket and make the hip more stable. There are a couple of things that can cause a torn labrum. If you have FAI (femoral acetabular impingement), that can pinch and tear the labrum. Or, just with wear and tear, the labrum can degenerate and tear. A torn labrum is a lot like a torn meniscal cartilage in the knee, and as such, can cause some chronic soreness and stiffness in the joint.
SI joint pain is usually a sharp pain in the posterior aspect of the very low back, basically where the little dimples are on the low back. It can sometimes be increased by doing a FABER maneuver with the leg on the same side. FABER stands for flexion, abduction, and external rotation. This is done usually, by the examiner moving the affected side leg up and then rotate it so that the ankle is on the other knee. Sort of like a figure-4 position, but with the patient lying down. This stresses the sacroiliac ligaments, and usually increases the patient's pain.
Inflammation of the SI joints can be easily picked up on a bone scan, which is the usually recommended study for SI joint arthropathy. In advanced stages, a CT scan will show degenerative changes within the SI joints.
Cortisone injections can be given for bursitis, especially around the greater trochanter. But, as you state, you have to have a discrete indication so that it can be placed in the correct place (either by examination or a study). Injections into the hip joint are usually avoided as long as the patient has fairly normal articular cartilage. These injections are sometimes done when a patient has advanced osteoarthritic changes in the hip and is looking at a total hip replacement. Then a cortisone injection may delay having to have a total hip. But, cortisone can damage normal cartilage, so its use in normal joints should be very limited. If the SI joint is found to be the cause, and everything else fails, sometimes an SI injection is done under x-ray guidance.
It sounds like you have tried almost all of the usual treatment. But, still don't really have a firm diagnosis. Have you seen an orthopedic hip surgeon? If not, that is probably the way you should go.
FAI is a fairly new disorder and is not known about by a lot of non-orthopedic physicians. If a physician is not well versed in it, he/she will not pick it up on the hip x-rays. Another reason to see a hip surgeon.
Hope you figure out what exactly is causing your problems, and it can be treated. Unfortunately, sometimes patients have to give up running and impact activities, and switch to things like biking and swimming. Hopefully, you won't have to do that completely.
Good luck.
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