I am by no means a doctor, but I do have a strong Kinesiology background and have had 3 shoulder surgeries in my life so I do know a bit about it.
The shoulder is easily the most complex joint in your body. The sheer amount of axis it moves on is incredible, and unfortunately makes it extremely easy to hurt.
With your specific injury, it all depends on what the doc found in your CT scan. First, if the tendon/ligament is torn, the only way to actually repair it is surgery. It won't grow back together by itself. Most doctors will recommend PT first, but all that really does is strengthen the muscles around it to relieve some of the pressure, and therefore pain. Depending on the ligament or tendon will change the difficulty of the surgery and recovery. Rotator cuff isn't horrible, either is a bicep tendon tear. But very rarely is it simply a tear of only one. Most times you will be looking at something like a SLAP tear to the glenoid labrum (which fucking sucks by the way) along with a fray to the bicep tendon.
Now, if it is simply inflamed, that's good news. It can be treated with antiinflammatories, but the only thing that will really heal it is rest. The biggest problem with the injections is that while they help in the short term, the stuff they use (normally cortisone or toridol) actually deteriorates the tendon/ligament in the long term.
I would offer caution to removing anything. Remember, everything in your body is there for a reason, and completely removing something will always have consequences, even if they are relatively small.
Sorry for the long-winded response, this is just kinda right in my wheelhouse.