The GOP did/does, they've repeatedly talked about health insurance competition across state lines, like auto insurance. They've also talked about TORT reform which would put the onus on people filing bogus medical malpractice lawsuits, and also lower the insurance doctors are forced to carry to protect against such lawsuits. The doctors jack up their procedure prices to make up for the high cost of malpractice insurance.
The two things alone would go a long way to reduce the cost of healthcare. However, the Dems would have none of it b/c TORT reform affects their trial lawyer donors.