I started seeing a family doctor in 2018 for diabetes.   Family doctor looked at my blood tests results and said.... normal recommendation is to put you on a statin because of diabetes. However, your cholesterol numbers are low, and I am not going to recommend it.  So, he didn't put me on a statin.  

Family doctor retired.... end of 2023?. Started with a new family doctor in 2024. Young guy.  This winter I had a stroke.  Doctor (heart doctor?  neurologist?  not sure which specialist?) put me on 80 mg Atorvastatin.  That was March, maybe?  Had new blood test in early May.   Had visit with family doctor in May.  Family doctor reduced Atorvastatin to 20 mg.  Had another blood test in July.  LDL was really low.  I questioned family doctor why I am taking a statin if my cholesterol levels are so low.  I asked whether they can get too low. He said he was wondering the same questions.  He reduced my Atorvastatin down to 10 mg.   I have another visit next month. I am interested in whether he wants to keep me on 10mg dose, or drop it altogether.   We'll see.

As far as side effects, I can't notice that I have an side effects. However, if your cholesterol levels are good, I would question your doctor if a dose lower than 80 mg of Atorvastatin will be acceptable? As far as Rosuvastatin, I don't know anything about it.  If Rosuvastatin is a new drug, there may be a cost savings to going with Atorvastatin compared to the Rosevastatin.  

I believe doctors probably have preferences to which medicines they prescribe, and maybe this is what you are running into.  Kinda like crop advisors have preferences to which herbicides they prescribe......