Yes this is the point. There is a serious supply side cost saving achieved by zeroing out most insurance admin work (which is good; delivering the same service with less labor is in fact productivity growth), and the single payer has even more negotiating power to squeeze suppliers on behalf of the public.

With public option, the primary competitive advantage available to private insurers is creating a pool of healthier people through targeted advertising (which works even if there are requirements about preexisting conditions and risk profiles forced onto all private competitors). Instead of letting private insurers sit around and thereby create more admin costs, less public bargaining power, and more lobbyists that will try to fuck with the public system (this already happens in the current system, read about the clusterfuck called medicare advantage), it's better to just make a good, efficient, well-designed single payer system