Not really. The https://en.wikipedia.org/wiki/Diamond%E2%80%93Dybvig_model argues that a better way of preventing bank runs is *deposit insurance* backed by the government or central bank. Such insurance pays depositors all or part of their losses in the case of a bank run. If depositors know that they will get their money back even in case of a bank run, they have no reason to participate in a bank run.
So, not necessarily the government. In practice it's always one of the central banks, either US or Europe. In the US it's the Federal Deposit Insurance Corporation.