Rheumatologist are so busy now. Most will refuse to see you without a referral, and they refuse most referrals. My experience has been that they are so busy, that unless a primary care doc has successfully done all the workup, and hands them a patient who carries a rheumatologic diagnosis already, they will decline to see the patient, or wait to see them for 6+ months.

Please do not try to goop up a rheumatologist schedule for shoulder pain that already carries a solid diagnosis of osteoarthritis.