I think there's a wide range of outcomes with frozen shoulder and it'd be hard for someone with a less serious version to understand why a doc would want to do surgery or not. I'm sure the docs look at X-rays and try to figure out what would be done but all that is way beyond me.
I had one cortisone shot that I felt "started" making things better but I got it at the height of the pain and as you note in the thawing stage things do get better on their own. The doc knew I was diabetic and made an effort to do the cortisone in a way to have minimal affect on my bg's and it seems to have worked - I saw no obvious effect on my bg's. At the same time, he was reluctant to do a cortisone shot more than once (I was still in some pain a month later and he pretty much told me "that's all I'm gonna get you kid").
In the past 10 months the sharp shooting pains (they were really bad, and could be triggered not just by hitting extreme of motion but just any jar to my body could set them off) have gone away and I have gone from maybe 60% range to nearly 95% range of motion in front of me and over my head. This is tracked by my PT. It's amazing, I can now reach straight up and touch my hands over my head with only mild discomfort - 10 months ago at this time I could barely extend my arm and raise it over my shoulder.
Some sources I read indicate that maybe lack of use of full range of motion, can contribute to frozen shoulder. If so, I wish I had been doing exercises etc. before the pain, not the PT after it. The link you have to Mayo clinic stuff is a good start, but you can also google "frozen shoulder exercises" and see a lot of the simple stretches that may help prevent loss of range of motion beforehand.