In fact, one of their rising stars has developed legislation along those lines:
In April, 137 Republicans voted in support of a GOP alternative budget. It didn’t generate a lot of attention, but the plan, drafted by the House Budget Committee’s Rep. Paul Ryan (R-Wis.) called for “replacing the traditional Medicare program with subsidies to help retirees enroll in private health care plans.”
Republicans hate Medicare. Period. They also hate Social Security and Medicaid.
137 Republicans. Members of the House. Some of them held town hall meetings loaded with old people screaming about keeping the government's hands off their Medicare. Who knew they agreed so thoroughly? Of course the old people were accidentally agreeing, but yeah.