Would it work if we privatized things like phone and internet but have the government set a price cap or a cap on the revenue that the company generates?
This stuff has more to do with monopoly than with private vs. public administration. The reason the private healthcare system is so much costlier here is that because the insurance companies are not required to cover everyone, they have a huge incentive to spend their resources avoiding givign coverage/care to the neediest people. The result being that all the money is spent on screening, and the needy people end up getting shafted. What fixes this is the granting of a monopoly: by requiring a single insurer to cover everyone, you eliminate the whole issue, and they are left to focus on improving the efficiency of actually delivering needed care. Of course, it may not be a good idea to grant a monopoly to a private company, but the point is that the difference has to do with the monopolistic nature of these other systems, rather than with their public nature as such.