A nice idea, as you say let's get some confusion!. When it comes to joint integrity: the joints blood supply is most likely estabished by the orthosympathetic pathways thus it could well have nothing to do with C1-C7 since these pathways come from the Thoracic area. Mobilisations of e.g. T1-4 can have dramatic effect on the pain as well as mobilisation rib1-4 9 (in between lie the orthosympathetis nodules) and in case the frozen shoulder is not well established the ROM as well. One should though ask why the shoulder and not other joints? Because the arm has it's nerve connection e.g. dermatomes, sclerotomes from C3-T2 and only the shoulder is affected? What makes the shoulder so special? the golfball on the T: an extremely unstable joint with via the Scapula only 1 bony connection the Clavicula.
Frozen shoulder does show intra articular changes as shrinking of capsula which is not explained by the explanation given by gerry the neck. Taking in account the possibility of reduction of bloodflow due to orthosympathetic changes here might lie also a clue. again why only the shoulder and no other joint?