I wouldn't expect competition among insurance companies to lower prices by very much. Insurance companies are already constrained by law as to how much profit they can make and there is only so much overhead you can remove.

In terms of competition, the real problem is the artificially low numbers of health care workers and facilities. The Drs are limited by the low number of medical schools, etc and the number of facilities are limited by requiring Certificates of Need, etc.

First remove these barriers to entry and require transparent pricing. Then... maybe competition would result in lower prices.