There could be a variety of issues but likely you have nailed it on the head with range of motion in the shoulder capsule / back and I would further expand that to relative strength of back, shoulder, hips and torso.
Of course, if you want to "bridge" you need to, well...practice bridging more, but I think more importantly you need to look at the attributes on bridging, consider what your training lacks and build it in. Typically few exercise enthusiasts / athletes today make use of movements that enhance dynamic range of motion and tumbling exercise but given your goals of actual performance (something rare these days), they should be apart of your training regime.
For your reference, within an article I published here known as the "Renegade Challenges" I enclosed a photo of one of my athletes/coaches going from a bridge to a complete standing position, which is an extraordinary display of athleticism. You may want to take a look at for your reference.
Best of luck and let me know how I can help.