Caps on damage awards have almost zero effect on rates. The insurance companies are sticking it to Doctor's, not the lawyers.
Neurosurgeons aren't refusing to perform spinal surgies because of premiums. It is because they are so likely to get sued if something goes wrong on a high-risk procedure.
You are correct, to a degree...
Yes, the insurance carriers ARE sticking it to Doctor's, and the reasons for that ARE varied and include that the markets right now are not providing the insurer's needed rate of return on invested premiums. That will change over time.
But it is also due to the fact that insurer's CAN stick it to Doctors because the number of insurers writing malpractice insurance has significantly declined, as some that wrote insurance with reasonable margins saw their investment income dry up and faced an uncertain risk due to moronic jury awards. In effect, the lawyers are sticking it to the insurers, and then the insurers turn around and either stick it to the Doctors or recently, leave the business.
Capping damage awards will to some degree improve the environment such that more insurer's may be willing to underwrite policies and an increase in the number of underwriters will increase competition and reduce rates.